Calculus Examples

$q = \cos (\frac{t}{\sqrt{t + 4}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{t + 4}$ as ${(t + 4)}^{\frac{1}{2}}$ .

$q = \cos (\frac{t}{{(t + 4)}^{\frac{1}{2}}})$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d q} (q) = \frac{d}{d q} (\cos (\frac{t}{{(t + 4)}^{\frac{1}{2}}}))$

Step 3

Differentiate using the Power Rule which states that $\frac{d}{d q} [q^{n}]$ is $n q^{n - 1}$ where $n = 1$ .

$1$

Step 4

Differentiate the right side of the equation.

Tap for more steps...

Step 4.1

Differentiate using the chain rule, which states that $\frac{d}{d q} [f (g (q))]$ is $f' (g (q)) g' (q)$ where $f (q) = \cos (q)$ and $g (q) = \frac{t}{{(t + 4)}^{\frac{1}{2}}}$ .

Tap for more steps...

Step 4.1.1

To apply the Chain Rule, set $u_{1}$ as $\frac{t}{{(t + 4)}^{\frac{1}{2}}}$ .

$\frac{d}{d u_{1}} [\cos (u_{1})] \frac{d}{d q} [\frac{t}{{(t + 4)}^{\frac{1}{2}}}]$

Step 4.1.2

The derivative of $\cos (u_{1})$ with respect to $u_{1}$ is $- \sin (u_{1})$ .

$- \sin (u_{1}) \frac{d}{d q} [\frac{t}{{(t + 4)}^{\frac{1}{2}}}]$

Step 4.1.3

Replace all occurrences of $u_{1}$ with $\frac{t}{{(t + 4)}^{\frac{1}{2}}}$ .

$- \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) \frac{d}{d q} [\frac{t}{{(t + 4)}^{\frac{1}{2}}}]$

Step 4.2

Differentiate using the Quotient Rule which states that $\frac{d}{d q} [\frac{f (q)}{g (q)}]$ is $\frac{g (q) \frac{d}{d q} [f (q)] - f (q) \frac{d}{d q} [g (q)]}{{g (q)}^{2}}$ where $f (q) = t$ and $g (q) = {(t + 4)}^{\frac{1}{2}}$ .

$- \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) \frac{{(t + 4)}^{\frac{1}{2}} \frac{d}{d q} [t] - t \frac{d}{d q} [{(t + 4)}^{\frac{1}{2}}]}{{({(t + 4)}^{\frac{1}{2}})}^{2}}$

Step 4.3

Multiply the exponents in ${({(t + 4)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 4.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) \frac{{(t + 4)}^{\frac{1}{2}} \frac{d}{d q} [t] - t \frac{d}{d q} [{(t + 4)}^{\frac{1}{2}}]}{{(t + 4)}^{\frac{1}{2} \cdot 2}}$

Step 4.3.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.3.2.1

Cancel the common factor.

$- \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) \frac{{(t + 4)}^{\frac{1}{2}} \frac{d}{d q} [t] - t \frac{d}{d q} [{(t + 4)}^{\frac{1}{2}}]}{{(t + 4)}^{\frac{1}{2} \cdot 2}}$

Step 4.3.2.2

Rewrite the expression.

$- \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) \frac{{(t + 4)}^{\frac{1}{2}} \frac{d}{d q} [t] - t \frac{d}{d q} [{(t + 4)}^{\frac{1}{2}}]}{{(t + 4)}^{1}}$

Step 4.4

Simplify.

$- \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) \frac{{(t + 4)}^{\frac{1}{2}} \frac{d}{d q} [t] - t \frac{d}{d q} [{(t + 4)}^{\frac{1}{2}}]}{t + 4}$

Step 4.5

Rewrite $\frac{d}{d q} [t]$ as $t'$ .

$- \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) \frac{{(t + 4)}^{\frac{1}{2}} t' - t \frac{d}{d q} [{(t + 4)}^{\frac{1}{2}}]}{t + 4}$

Step 4.6

Differentiate using the chain rule, which states that $\frac{d}{d q} [f (g (q))]$ is $f' (g (q)) g' (q)$ where $f (q) = q^{\frac{1}{2}}$ and $g (q) = t + 4$ .

Tap for more steps...

Step 4.6.1

To apply the Chain Rule, set $u_{2}$ as $t + 4$ .

$- \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) \frac{{(t + 4)}^{\frac{1}{2}} t' - t (\frac{d}{d u_{2}} [{u_{2}}^{\frac{1}{2}}] \frac{d}{d q} [t + 4])}{t + 4}$

Step 4.6.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = \frac{1}{2}$ .

$- \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) \frac{{(t + 4)}^{\frac{1}{2}} t' - t (\frac{1}{2} {u_{2}}^{\frac{1}{2} - 1} \frac{d}{d q} [t + 4])}{t + 4}$

Step 4.6.3

Replace all occurrences of $u_{2}$ with $t + 4$ .

$- \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) \frac{{(t + 4)}^{\frac{1}{2}} t' - t (\frac{1}{2} {(t + 4)}^{\frac{1}{2} - 1} \frac{d}{d q} [t + 4])}{t + 4}$

Step 4.7

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) \frac{{(t + 4)}^{\frac{1}{2}} t' - t (\frac{1}{2} {(t + 4)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d q} [t + 4])}{t + 4}$

Step 4.8

Combine $- 1$ and $\frac{2}{2}$ .

$- \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) \frac{{(t + 4)}^{\frac{1}{2}} t' - t (\frac{1}{2} {(t + 4)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d q} [t + 4])}{t + 4}$

Step 4.9

Combine the numerators over the common denominator.

$- \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) \frac{{(t + 4)}^{\frac{1}{2}} t' - t (\frac{1}{2} {(t + 4)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d q} [t + 4])}{t + 4}$

Step 4.10

Simplify the numerator.

Tap for more steps...

Step 4.10.1

Multiply $- 1$ by $2$ .

$- \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) \frac{{(t + 4)}^{\frac{1}{2}} t' - t (\frac{1}{2} {(t + 4)}^{\frac{1 - 2}{2}} \frac{d}{d q} [t + 4])}{t + 4}$

Step 4.10.2

Subtract $2$ from $1$ .

$- \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) \frac{{(t + 4)}^{\frac{1}{2}} t' - t (\frac{1}{2} {(t + 4)}^{\frac{- 1}{2}} \frac{d}{d q} [t + 4])}{t + 4}$

Step 4.11

Combine fractions.

Tap for more steps...

Step 4.11.1

Move the negative in front of the fraction.

$- \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) \frac{{(t + 4)}^{\frac{1}{2}} t' - t (\frac{1}{2} {(t + 4)}^{- \frac{1}{2}} \frac{d}{d q} [t + 4])}{t + 4}$

Step 4.11.2

Combine $\frac{1}{2}$ and ${(t + 4)}^{- \frac{1}{2}}$ .

$- \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) \frac{{(t + 4)}^{\frac{1}{2}} t' - t (\frac{{(t + 4)}^{- \frac{1}{2}}}{2} \frac{d}{d q} [t + 4])}{t + 4}$

Step 4.11.3

Move ${(t + 4)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) \frac{{(t + 4)}^{\frac{1}{2}} t' - t (\frac{1}{2 {(t + 4)}^{\frac{1}{2}}} \frac{d}{d q} [t + 4])}{t + 4}$

Step 4.11.4

Combine $\frac{1}{2 {(t + 4)}^{\frac{1}{2}}}$ and $t$ .

$- \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) \frac{{(t + 4)}^{\frac{1}{2}} t' - \frac{t}{2 {(t + 4)}^{\frac{1}{2}}} \frac{d}{d q} [t + 4]}{t + 4}$

Step 4.12

By the Sum Rule, the derivative of $t + 4$ with respect to $q$ is $\frac{d}{d q} [t] + \frac{d}{d q} [4]$ .

$- \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) \frac{{(t + 4)}^{\frac{1}{2}} t' - \frac{t}{2 {(t + 4)}^{\frac{1}{2}}} (\frac{d}{d q} [t] + \frac{d}{d q} [4])}{t + 4}$

Step 4.13

Rewrite $\frac{d}{d q} [t]$ as $t'$ .

$- \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) \frac{{(t + 4)}^{\frac{1}{2}} t' - \frac{t}{2 {(t + 4)}^{\frac{1}{2}}} (t' + \frac{d}{d q} [4])}{t + 4}$

Step 4.14

Since $4$ is constant with respect to $q$ , the derivative of $4$ with respect to $q$ is $0$ .

$- \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) \frac{{(t + 4)}^{\frac{1}{2}} t' - \frac{t}{2 {(t + 4)}^{\frac{1}{2}}} (t' + 0)}{t + 4}$

Step 4.15

Combine fractions.

Tap for more steps...

Step 4.15.1

Add $t'$ and $0$ .

$- \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) \frac{{(t + 4)}^{\frac{1}{2}} t' - \frac{t}{2 {(t + 4)}^{\frac{1}{2}}} t'}{t + 4}$

Step 4.15.2

Combine $t'$ and $\frac{t}{2 {(t + 4)}^{\frac{1}{2}}}$ .

$- \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) \frac{{(t + 4)}^{\frac{1}{2}} t' - \frac{t' t}{2 {(t + 4)}^{\frac{1}{2}}}}{t + 4}$

Step 4.15.3

Combine $\frac{{(t + 4)}^{\frac{1}{2}} t' - \frac{t' t}{2 {(t + 4)}^{\frac{1}{2}}}}{t + 4}$ and $\sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}})$ .

$- \frac{({(t + 4)}^{\frac{1}{2}} t' - \frac{t' t}{2 {(t + 4)}^{\frac{1}{2}}}) \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}})}{t + 4}$

Step 4.16

Simplify.

Tap for more steps...

Step 4.16.1

Simplify the numerator.

Tap for more steps...

Step 4.16.1.1

Factor $t'$ out of ${(t + 4)}^{\frac{1}{2}} t' - \frac{t' t}{2 {(t + 4)}^{\frac{1}{2}}}$ .

Tap for more steps...

Step 4.16.1.1.1

Factor $t'$ out of ${(t + 4)}^{\frac{1}{2}} t'$ .

$- \frac{(t' {(t + 4)}^{\frac{1}{2}} - \frac{t' t}{2 {(t + 4)}^{\frac{1}{2}}}) \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}})}{t + 4}$

Step 4.16.1.1.2

Factor $t'$ out of $- \frac{t' t}{2 {(t + 4)}^{\frac{1}{2}}}$ .

$- \frac{(t' {(t + 4)}^{\frac{1}{2}} + t' (- \frac{t}{2 {(t + 4)}^{\frac{1}{2}}})) \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}})}{t + 4}$

Step 4.16.1.1.3

Factor $t'$ out of $t' {(t + 4)}^{\frac{1}{2}} + t' (- \frac{t}{2 {(t + 4)}^{\frac{1}{2}}})$ .

$- \frac{t' ({(t + 4)}^{\frac{1}{2}} - \frac{t}{2 {(t + 4)}^{\frac{1}{2}}}) \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}})}{t + 4}$

Step 4.16.1.2

To write ${(t + 4)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2 {(t + 4)}^{\frac{1}{2}}}{2 {(t + 4)}^{\frac{1}{2}}}$ .

$- \frac{t' ({(t + 4)}^{\frac{1}{2}} \cdot \frac{2 {(t + 4)}^{\frac{1}{2}}}{2 {(t + 4)}^{\frac{1}{2}}} - \frac{t}{2 {(t + 4)}^{\frac{1}{2}}}) \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}})}{t + 4}$

Step 4.16.1.3

Combine ${(t + 4)}^{\frac{1}{2}}$ and $\frac{2 {(t + 4)}^{\frac{1}{2}}}{2 {(t + 4)}^{\frac{1}{2}}}$ .

$- \frac{t' (\frac{{(t + 4)}^{\frac{1}{2}} (2 {(t + 4)}^{\frac{1}{2}})}{2 {(t + 4)}^{\frac{1}{2}}} - \frac{t}{2 {(t + 4)}^{\frac{1}{2}}}) \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}})}{t + 4}$

Step 4.16.1.4

Combine the numerators over the common denominator.

$- \frac{t' \frac{{(t + 4)}^{\frac{1}{2}} (2 {(t + 4)}^{\frac{1}{2}}) - t}{2 {(t + 4)}^{\frac{1}{2}}} \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}})}{t + 4}$

Step 4.16.1.5

Rewrite $\frac{{(t + 4)}^{\frac{1}{2}} (2 {(t + 4)}^{\frac{1}{2}}) - t}{2 {(t + 4)}^{\frac{1}{2}}}$ in a factored form.

Tap for more steps...

Step 4.16.1.5.1

Rewrite using the commutative property of multiplication.

$- \frac{t' \frac{2 {(t + 4)}^{\frac{1}{2}} {(t + 4)}^{\frac{1}{2}} - t}{2 {(t + 4)}^{\frac{1}{2}}} \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}})}{t + 4}$

Step 4.16.1.5.2

Multiply ${(t + 4)}^{\frac{1}{2}}$ by ${(t + 4)}^{\frac{1}{2}}$ by adding the exponents.

Tap for more steps...

Step 4.16.1.5.2.1

Move ${(t + 4)}^{\frac{1}{2}}$ .

$- \frac{t' \frac{2 ({(t + 4)}^{\frac{1}{2}} {(t + 4)}^{\frac{1}{2}}) - t}{2 {(t + 4)}^{\frac{1}{2}}} \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}})}{t + 4}$

Step 4.16.1.5.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{t' \frac{2 {(t + 4)}^{\frac{1}{2} + \frac{1}{2}} - t}{2 {(t + 4)}^{\frac{1}{2}}} \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}})}{t + 4}$

Step 4.16.1.5.2.3

Combine the numerators over the common denominator.

$- \frac{t' \frac{2 {(t + 4)}^{\frac{1 + 1}{2}} - t}{2 {(t + 4)}^{\frac{1}{2}}} \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}})}{t + 4}$

Step 4.16.1.5.2.4

Add $1$ and $1$ .

$- \frac{t' \frac{2 {(t + 4)}^{\frac{2}{2}} - t}{2 {(t + 4)}^{\frac{1}{2}}} \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}})}{t + 4}$

Step 4.16.1.5.2.5

Divide $2$ by $2$ .

$- \frac{t' \frac{2 {(t + 4)}^{1} - t}{2 {(t + 4)}^{\frac{1}{2}}} \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}})}{t + 4}$

Step 4.16.1.5.3

Simplify $2 {(t + 4)}^{1}$ .

$- \frac{t' \frac{2 (t + 4) - t}{2 {(t + 4)}^{\frac{1}{2}}} \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}})}{t + 4}$

Step 4.16.1.5.4

Apply the distributive property.

$- \frac{t' \frac{2 t + 2 \cdot 4 - t}{2 {(t + 4)}^{\frac{1}{2}}} \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}})}{t + 4}$

Step 4.16.1.5.5

Multiply $2$ by $4$ .

$- \frac{t' \frac{2 t + 8 - t}{2 {(t + 4)}^{\frac{1}{2}}} \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}})}{t + 4}$

Step 4.16.1.5.6

Subtract $t$ from $2 t$ .

$- \frac{t' \frac{t + 8}{2 {(t + 4)}^{\frac{1}{2}}} \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}})}{t + 4}$

Step 4.16.1.6

Combine exponents.

Tap for more steps...

Step 4.16.1.6.1

Combine $t'$ and $\frac{t + 8}{2 {(t + 4)}^{\frac{1}{2}}}$ .

$- \frac{\frac{t' (t + 8)}{2 {(t + 4)}^{\frac{1}{2}}} \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}})}{t + 4}$

Step 4.16.1.6.2

Combine $\frac{t' (t + 8)}{2 {(t + 4)}^{\frac{1}{2}}}$ and $\sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}})$ .

$- \frac{\frac{t' (t + 8) \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}})}{2 {(t + 4)}^{\frac{1}{2}}}}{t + 4}$

Step 4.16.2

Combine terms.

Tap for more steps...

Step 4.16.2.1

Rewrite $\frac{\frac{t' (t + 8) \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}})}{2 {(t + 4)}^{\frac{1}{2}}}}{t + 4}$ as a product.

$- (\frac{t' (t + 8) \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}})}{2 {(t + 4)}^{\frac{1}{2}}} \cdot \frac{1}{t + 4})$

Step 4.16.2.2

Multiply $\frac{t' (t + 8) \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}})}{2 {(t + 4)}^{\frac{1}{2}}}$ by $\frac{1}{t + 4}$ .

$- \frac{t' (t + 8) \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}})}{2 {(t + 4)}^{\frac{1}{2}} (t + 4)}$

Step 4.16.2.3

Raise $t + 4$ to the power of $1$ .

$- \frac{t' (t + 8) \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}})}{2 ({(t + 4)}^{1} {(t + 4)}^{\frac{1}{2}})}$

Step 4.16.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{t' (t + 8) \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}})}{2 {(t + 4)}^{1 + \frac{1}{2}}}$

Step 4.16.2.5

Write $1$ as a fraction with a common denominator.

$- \frac{t' (t + 8) \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}})}{2 {(t + 4)}^{\frac{2}{2} + \frac{1}{2}}}$

Step 4.16.2.6

Combine the numerators over the common denominator.

$- \frac{t' (t + 8) \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}})}{2 {(t + 4)}^{\frac{2 + 1}{2}}}$

Step 4.16.2.7

Add $2$ and $1$ .

$- \frac{t' (t + 8) \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}})}{2 {(t + 4)}^{\frac{3}{2}}}$

Step 4.16.3

Reorder factors in $- \frac{t' (t + 8) \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}})}{2 {(t + 4)}^{\frac{3}{2}}}$ .

$- \frac{\sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) (t + 8) t'}{2 {(t + 4)}^{\frac{3}{2}}}$

Step 5

Reform the equation by setting the left side equal to the right side.

$1 = - \frac{\sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) (t + 8) t'}{2 {(t + 4)}^{\frac{3}{2}}}$

Step 6

Solve for $t'$ .

Tap for more steps...

Step 6.1

Rewrite the equation as $- \frac{\sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) (t + 8) t'}{2 {(t + 4)}^{\frac{3}{2}}} = 1$ .

$- \frac{\sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) (t + 8) t'}{2 {(t + 4)}^{\frac{3}{2}}} = 1$

Step 6.2

Divide each term in $- \frac{\sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) (t + 8) t'}{2 {(t + 4)}^{\frac{3}{2}}} = 1$ by $- 1$ and simplify.

Tap for more steps...

Step 6.2.1

Divide each term in $- \frac{\sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) (t + 8) t'}{2 {(t + 4)}^{\frac{3}{2}}} = 1$ by $- 1$ .

$\frac{- \frac{\sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) (t + 8) t'}{2 {(t + 4)}^{\frac{3}{2}}}}{- 1} = \frac{1}{- 1}$

Step 6.2.2

Simplify the left side.

Tap for more steps...

Step 6.2.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{\sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) (t + 8) t'}{2 {(t + 4)}^{\frac{3}{2}}}}{1} = \frac{1}{- 1}$

Step 6.2.2.2

Simplify the expression.

Tap for more steps...

Step 6.2.2.2.1

Divide $\frac{\sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) (t + 8) t'}{2 {(t + 4)}^{\frac{3}{2}}}$ by $1$ .

$\frac{\sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) (t + 8) t'}{2 {(t + 4)}^{\frac{3}{2}}} = \frac{1}{- 1}$

Step 6.2.2.2.2

Reorder factors in $\frac{\sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) (t + 8) t'}{2 {(t + 4)}^{\frac{3}{2}}}$ .

$\frac{t' \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) (t + 8)}{2 {(t + 4)}^{\frac{3}{2}}} = \frac{1}{- 1}$

Step 6.2.3

Simplify the right side.

Tap for more steps...

Step 6.2.3.1

Divide $1$ by $- 1$ .

$\frac{t' \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) (t + 8)}{2 {(t + 4)}^{\frac{3}{2}}} = - 1$

Step 6.3

Multiply both sides by $2 {(t + 4)}^{\frac{3}{2}}$ .

$\frac{t' \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) (t + 8)}{2 {(t + 4)}^{\frac{3}{2}}} (2 {(t + 4)}^{\frac{3}{2}}) = - (2 {(t + 4)}^{\frac{3}{2}})$

Step 6.4

Simplify.

Tap for more steps...

Step 6.4.1

Simplify the left side.

Tap for more steps...

Step 6.4.1.1

Simplify $\frac{t' \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) (t + 8)}{2 {(t + 4)}^{\frac{3}{2}}} (2 {(t + 4)}^{\frac{3}{2}})$ .

Tap for more steps...

Step 6.4.1.1.1

Simplify terms.

Tap for more steps...

Step 6.4.1.1.1.1

Rewrite using the commutative property of multiplication.

$2 \frac{t' \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) (t + 8)}{2 {(t + 4)}^{\frac{3}{2}}} {(t + 4)}^{\frac{3}{2}} = - (2 {(t + 4)}^{\frac{3}{2}})$

Step 6.4.1.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.4.1.1.1.2.1

Cancel the common factor.

$2 \frac{t' \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) (t + 8)}{2 {(t + 4)}^{\frac{3}{2}}} {(t + 4)}^{\frac{3}{2}} = - (2 {(t + 4)}^{\frac{3}{2}})$

Step 6.4.1.1.1.2.2

Rewrite the expression.

$\frac{t' \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) (t + 8)}{{(t + 4)}^{\frac{3}{2}}} {(t + 4)}^{\frac{3}{2}} = - (2 {(t + 4)}^{\frac{3}{2}})$

Step 6.4.1.1.1.3

Cancel the common factor of ${(t + 4)}^{\frac{3}{2}}$ .

Tap for more steps...

Step 6.4.1.1.1.3.1

Cancel the common factor.

$\frac{t' \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) (t + 8)}{{(t + 4)}^{\frac{3}{2}}} {(t + 4)}^{\frac{3}{2}} = - (2 {(t + 4)}^{\frac{3}{2}})$

Step 6.4.1.1.1.3.2

Rewrite the expression.

$t' \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) (t + 8) = - (2 {(t + 4)}^{\frac{3}{2}})$

Step 6.4.1.1.1.4

Apply the distributive property.

$t' \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) t + t' \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) \cdot 8 = - (2 {(t + 4)}^{\frac{3}{2}})$

Step 6.4.1.1.1.5

Move $8$ to the left of $t' \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}})$ .

$t' \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) t + 8 \cdot (t' \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}})) = - (2 {(t + 4)}^{\frac{3}{2}})$

Step 6.4.1.1.2

Remove parentheses.

$t' \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) t + 8 t' \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) = - (2 {(t + 4)}^{\frac{3}{2}})$

Step 6.4.1.1.3

Simplify the expression.

Tap for more steps...

Step 6.4.1.1.3.1

Reorder factors in $t' \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) t + 8 t' \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}})$ .

$t' t \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) + 8 t' \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) = - (2 {(t + 4)}^{\frac{3}{2}})$

Step 6.4.1.1.3.2

Reorder $t'$ and $t$ .

$t t' \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) + 8 t' \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) = - (2 {(t + 4)}^{\frac{3}{2}})$

Step 6.4.2

Simplify the right side.

Tap for more steps...

Step 6.4.2.1

Multiply $2$ by $- 1$ .

$t t' \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) + 8 t' \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) = - 2 {(t + 4)}^{\frac{3}{2}}$

Step 6.5

Solve for $t'$ .

Tap for more steps...

Step 6.5.1

Factor $t' \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}})$ out of $t t' \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) + 8 t' \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}})$ .

Tap for more steps...

Step 6.5.1.1

Factor $t' \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}})$ out of $t t' \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}})$ .

$t' \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) t + 8 t' \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) = - 2 {(t + 4)}^{\frac{3}{2}}$

Step 6.5.1.2

Factor $t' \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}})$ out of $8 t' \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}})$ .

$t' \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) t + t' \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) \cdot 8 = - 2 {(t + 4)}^{\frac{3}{2}}$

Step 6.5.1.3

Factor $t' \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}})$ out of $t' \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) t + t' \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) \cdot 8$ .

$t' \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) (t + 8) = - 2 {(t + 4)}^{\frac{3}{2}}$

Step 6.5.2

Divide each term in $t' \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) (t + 8) = - 2 {(t + 4)}^{\frac{3}{2}}$ by $\sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) (t + 8)$ and simplify.

Tap for more steps...

Step 6.5.2.1

Divide each term in $t' \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) (t + 8) = - 2 {(t + 4)}^{\frac{3}{2}}$ by $\sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) (t + 8)$ .

$\frac{t' \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) (t + 8)}{\sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) (t + 8)} = \frac{- 2 {(t + 4)}^{\frac{3}{2}}}{\sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) (t + 8)}$

Step 6.5.2.2

Simplify the left side.

Tap for more steps...

Step 6.5.2.2.1

Cancel the common factor.

$\frac{t' \sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) (t + 8)}{\sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) (t + 8)} = \frac{- 2 {(t + 4)}^{\frac{3}{2}}}{\sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) (t + 8)}$

Step 6.5.2.2.2

Rewrite the expression.

$\frac{t' (t + 8)}{t + 8} = \frac{- 2 {(t + 4)}^{\frac{3}{2}}}{\sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) (t + 8)}$

Step 6.5.2.2.3

Cancel the common factor of $t + 8$ .

Tap for more steps...

Step 6.5.2.2.3.1

Cancel the common factor.

$\frac{t' (t + 8)}{t + 8} = \frac{- 2 {(t + 4)}^{\frac{3}{2}}}{\sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) (t + 8)}$

Step 6.5.2.2.3.2

Divide $t'$ by $1$ .

$t' = \frac{- 2 {(t + 4)}^{\frac{3}{2}}}{\sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) (t + 8)}$

Step 6.5.2.3

Simplify the right side.

Tap for more steps...

Step 6.5.2.3.1

Move the negative in front of the fraction.

$t' = - \frac{2 {(t + 4)}^{\frac{3}{2}}}{\sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) (t + 8)}$

Step 7

Replace $t'$ with $\frac{d t}{d q}$ .

$\frac{d t}{d q} = - \frac{2 {(t + 4)}^{\frac{3}{2}}}{\sin (\frac{t}{{(t + 4)}^{\frac{1}{2}}}) (t + 8)}$