Calculus Examples

$q = \cos (\frac{t}{\sqrt{t + 5}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{t + 5}$ as ${(t + 5)}^{\frac{1}{2}}$ .

$q = \cos (\frac{t}{{(t + 5)}^{\frac{1}{2}}})$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d t} (q) = \frac{d}{d t} (\cos (\frac{t}{{(t + 5)}^{\frac{1}{2}}}))$

Step 3

The derivative of $q$ with respect to $t$ is $q'$ .

$q'$

Step 4

Differentiate the right side of the equation.

Tap for more steps...

Step 4.1

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = \cos (t)$ and $g (t) = \frac{t}{{(t + 5)}^{\frac{1}{2}}}$ .

Tap for more steps...

Step 4.1.1

To apply the Chain Rule, set $u_{1}$ as $\frac{t}{{(t + 5)}^{\frac{1}{2}}}$ .

$\frac{d}{d u_{1}} [\cos (u_{1})] \frac{d}{d t} [\frac{t}{{(t + 5)}^{\frac{1}{2}}}]$

Step 4.1.2

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

$- \sin (u_{1}) \frac{d}{d t} [\frac{t}{{(t + 5)}^{\frac{1}{2}}}]$

Step 4.1.3

Replace all occurrences of $u_{1}$ with $\frac{t}{{(t + 5)}^{\frac{1}{2}}}$ .

$- \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \frac{d}{d t} [\frac{t}{{(t + 5)}^{\frac{1}{2}}}]$

Step 4.2

Differentiate using the Quotient Rule which states that $\frac{d}{d t} [\frac{f (t)}{g (t)}]$ is $\frac{g (t) \frac{d}{d t} [f (t)] - f (t) \frac{d}{d t} [g (t)]}{{g (t)}^{2}}$ where $f (t) = t$ and $g (t) = {(t + 5)}^{\frac{1}{2}}$ .

$- \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \frac{{(t + 5)}^{\frac{1}{2}} \frac{d}{d t} [t] - t \frac{d}{d t} [{(t + 5)}^{\frac{1}{2}}]}{{({(t + 5)}^{\frac{1}{2}})}^{2}}$

Step 4.3

Multiply the exponents in ${({(t + 5)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 4.3.1

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

$- \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \frac{{(t + 5)}^{\frac{1}{2}} \frac{d}{d t} [t] - t \frac{d}{d t} [{(t + 5)}^{\frac{1}{2}}]}{{(t + 5)}^{\frac{1}{2} \cdot 2}}$

Step 4.3.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.3.2.1

Cancel the common factor.

$- \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \frac{{(t + 5)}^{\frac{1}{2}} \frac{d}{d t} [t] - t \frac{d}{d t} [{(t + 5)}^{\frac{1}{2}}]}{{(t + 5)}^{\frac{1}{2} \cdot 2}}$

Step 4.3.2.2

Rewrite the expression.

$- \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \frac{{(t + 5)}^{\frac{1}{2}} \frac{d}{d t} [t] - t \frac{d}{d t} [{(t + 5)}^{\frac{1}{2}}]}{{(t + 5)}^{1}}$

Step 4.4

Simplify.

$- \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \frac{{(t + 5)}^{\frac{1}{2}} \frac{d}{d t} [t] - t \frac{d}{d t} [{(t + 5)}^{\frac{1}{2}}]}{t + 5}$

Step 4.5

Differentiate using the Power Rule.

Tap for more steps...

Step 4.5.1

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

$- \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \frac{{(t + 5)}^{\frac{1}{2}} \cdot 1 - t \frac{d}{d t} [{(t + 5)}^{\frac{1}{2}}]}{t + 5}$

Step 4.5.2

Multiply ${(t + 5)}^{\frac{1}{2}}$ by $1$ .

$- \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \frac{{(t + 5)}^{\frac{1}{2}} - t \frac{d}{d t} [{(t + 5)}^{\frac{1}{2}}]}{t + 5}$

Step 4.6

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{\frac{1}{2}}$ and $g (t) = t + 5$ .

Tap for more steps...

Step 4.6.1

To apply the Chain Rule, set $u_{2}$ as $t + 5$ .

$- \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \frac{{(t + 5)}^{\frac{1}{2}} - t (\frac{d}{d u_{2}} [{u_{2}}^{\frac{1}{2}}] \frac{d}{d t} [t + 5])}{t + 5}$

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 $n = \frac{1}{2}$ .

$- \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \frac{{(t + 5)}^{\frac{1}{2}} - t (\frac{1}{2} {u_{2}}^{\frac{1}{2} - 1} \frac{d}{d t} [t + 5])}{t + 5}$

Step 4.6.3

Replace all occurrences of $u_{2}$ with $t + 5$ .

$- \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \frac{{(t + 5)}^{\frac{1}{2}} - t (\frac{1}{2} {(t + 5)}^{\frac{1}{2} - 1} \frac{d}{d t} [t + 5])}{t + 5}$

Step 4.7

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

$- \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \frac{{(t + 5)}^{\frac{1}{2}} - t (\frac{1}{2} {(t + 5)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d t} [t + 5])}{t + 5}$

Step 4.8

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

$- \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \frac{{(t + 5)}^{\frac{1}{2}} - t (\frac{1}{2} {(t + 5)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d t} [t + 5])}{t + 5}$

Step 4.9

Combine the numerators over the common denominator.

$- \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \frac{{(t + 5)}^{\frac{1}{2}} - t (\frac{1}{2} {(t + 5)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d t} [t + 5])}{t + 5}$

Step 4.10

Simplify the numerator.

Tap for more steps...

Step 4.10.1

Multiply $- 1$ by $2$ .

$- \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \frac{{(t + 5)}^{\frac{1}{2}} - t (\frac{1}{2} {(t + 5)}^{\frac{1 - 2}{2}} \frac{d}{d t} [t + 5])}{t + 5}$

Step 4.10.2

Subtract $2$ from $1$ .

$- \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \frac{{(t + 5)}^{\frac{1}{2}} - t (\frac{1}{2} {(t + 5)}^{\frac{- 1}{2}} \frac{d}{d t} [t + 5])}{t + 5}$

Step 4.11

Combine fractions.

Tap for more steps...

Step 4.11.1

Move the negative in front of the fraction.

$- \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \frac{{(t + 5)}^{\frac{1}{2}} - t (\frac{1}{2} {(t + 5)}^{- \frac{1}{2}} \frac{d}{d t} [t + 5])}{t + 5}$

Step 4.11.2

Combine $\frac{1}{2}$ and ${(t + 5)}^{- \frac{1}{2}}$ .

$- \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \frac{{(t + 5)}^{\frac{1}{2}} - t (\frac{{(t + 5)}^{- \frac{1}{2}}}{2} \frac{d}{d t} [t + 5])}{t + 5}$

Step 4.11.3

Move ${(t + 5)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \frac{{(t + 5)}^{\frac{1}{2}} - t (\frac{1}{2 {(t + 5)}^{\frac{1}{2}}} \frac{d}{d t} [t + 5])}{t + 5}$

Step 4.11.4

Combine $\frac{1}{2 {(t + 5)}^{\frac{1}{2}}}$ and $t$ .

$- \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \frac{{(t + 5)}^{\frac{1}{2}} - \frac{t}{2 {(t + 5)}^{\frac{1}{2}}} \frac{d}{d t} [t + 5]}{t + 5}$

Step 4.12

By the Sum Rule, the derivative of $t + 5$ with respect to $t$ is $\frac{d}{d t} [t] + \frac{d}{d t} [5]$ .

$- \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \frac{{(t + 5)}^{\frac{1}{2}} - \frac{t}{2 {(t + 5)}^{\frac{1}{2}}} (\frac{d}{d t} [t] + \frac{d}{d t} [5])}{t + 5}$

Step 4.13

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

$- \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \frac{{(t + 5)}^{\frac{1}{2}} - \frac{t}{2 {(t + 5)}^{\frac{1}{2}}} (1 + \frac{d}{d t} [5])}{t + 5}$

Step 4.14

Since $5$ is constant with respect to $t$ , the derivative of $5$ with respect to $t$ is $0$ .

$- \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \frac{{(t + 5)}^{\frac{1}{2}} - \frac{t}{2 {(t + 5)}^{\frac{1}{2}}} (1 + 0)}{t + 5}$

Step 4.15

Simplify the expression.

Tap for more steps...

Step 4.15.1

Add $1$ and $0$ .

$- \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \frac{{(t + 5)}^{\frac{1}{2}} - \frac{t}{2 {(t + 5)}^{\frac{1}{2}}} \cdot 1}{t + 5}$

Step 4.15.2

Multiply $- 1$ by $1$ .

$- \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \frac{{(t + 5)}^{\frac{1}{2}} - \frac{t}{2 {(t + 5)}^{\frac{1}{2}}}}{t + 5}$

Step 4.16

To write ${(t + 5)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2 {(t + 5)}^{\frac{1}{2}}}{2 {(t + 5)}^{\frac{1}{2}}}$ .

$- \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \frac{{(t + 5)}^{\frac{1}{2}} \cdot \frac{2 {(t + 5)}^{\frac{1}{2}}}{2 {(t + 5)}^{\frac{1}{2}}} - \frac{t}{2 {(t + 5)}^{\frac{1}{2}}}}{t + 5}$

Step 4.17

Combine ${(t + 5)}^{\frac{1}{2}}$ and $\frac{2 {(t + 5)}^{\frac{1}{2}}}{2 {(t + 5)}^{\frac{1}{2}}}$ .

$- \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \frac{\frac{{(t + 5)}^{\frac{1}{2}} (2 {(t + 5)}^{\frac{1}{2}})}{2 {(t + 5)}^{\frac{1}{2}}} - \frac{t}{2 {(t + 5)}^{\frac{1}{2}}}}{t + 5}$

Step 4.18

Combine the numerators over the common denominator.

$- \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \frac{\frac{{(t + 5)}^{\frac{1}{2}} (2 {(t + 5)}^{\frac{1}{2}}) - t}{2 {(t + 5)}^{\frac{1}{2}}}}{t + 5}$

Step 4.19

Multiply ${(t + 5)}^{\frac{1}{2}}$ by ${(t + 5)}^{\frac{1}{2}}$ by adding the exponents.

Tap for more steps...

Step 4.19.1

Move ${(t + 5)}^{\frac{1}{2}}$ .

$- \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \frac{\frac{{(t + 5)}^{\frac{1}{2}} {(t + 5)}^{\frac{1}{2}} \cdot 2 - t}{2 {(t + 5)}^{\frac{1}{2}}}}{t + 5}$

Step 4.19.2

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

$- \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \frac{\frac{{(t + 5)}^{\frac{1}{2} + \frac{1}{2}} \cdot 2 - t}{2 {(t + 5)}^{\frac{1}{2}}}}{t + 5}$

Step 4.19.3

Combine the numerators over the common denominator.

$- \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \frac{\frac{{(t + 5)}^{\frac{1 + 1}{2}} \cdot 2 - t}{2 {(t + 5)}^{\frac{1}{2}}}}{t + 5}$

Step 4.19.4

Add $1$ and $1$ .

$- \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \frac{\frac{{(t + 5)}^{\frac{2}{2}} \cdot 2 - t}{2 {(t + 5)}^{\frac{1}{2}}}}{t + 5}$

Step 4.19.5

Divide $2$ by $2$ .

$- \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \frac{\frac{{(t + 5)}^{1} \cdot 2 - t}{2 {(t + 5)}^{\frac{1}{2}}}}{t + 5}$

Step 4.20

Simplify ${(t + 5)}^{1} \cdot 2$ .

$- \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \frac{\frac{(t + 5) \cdot 2 - t}{2 {(t + 5)}^{\frac{1}{2}}}}{t + 5}$

Step 4.21

Move $2$ to the left of $t + 5$ .

$- \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \frac{\frac{2 \cdot (t + 5) - t}{2 {(t + 5)}^{\frac{1}{2}}}}{t + 5}$

Step 4.22

Rewrite $\frac{\frac{2 (t + 5) - t}{2 {(t + 5)}^{\frac{1}{2}}}}{t + 5}$ as a product.

$- \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) (\frac{2 (t + 5) - t}{2 {(t + 5)}^{\frac{1}{2}}} \cdot \frac{1}{t + 5})$

Step 4.23

Multiply $\frac{2 (t + 5) - t}{2 {(t + 5)}^{\frac{1}{2}}}$ by $\frac{1}{t + 5}$ .

$- \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \frac{2 (t + 5) - t}{2 {(t + 5)}^{\frac{1}{2}} (t + 5)}$

Step 4.24

Raise $t + 5$ to the power of $1$ .

$- \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \frac{2 (t + 5) - t}{2 ({(t + 5)}^{1} {(t + 5)}^{\frac{1}{2}})}$

Step 4.25

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

$- \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \frac{2 (t + 5) - t}{2 {(t + 5)}^{1 + \frac{1}{2}}}$

Step 4.26

Simplify the expression.

Tap for more steps...

Step 4.26.1

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

$- \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \frac{2 (t + 5) - t}{2 {(t + 5)}^{\frac{2}{2} + \frac{1}{2}}}$

Step 4.26.2

Combine the numerators over the common denominator.

$- \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \frac{2 (t + 5) - t}{2 {(t + 5)}^{\frac{2 + 1}{2}}}$

Step 4.26.3

Add $2$ and $1$ .

$- \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \frac{2 (t + 5) - t}{2 {(t + 5)}^{\frac{3}{2}}}$

Step 4.27

Combine $\frac{2 (t + 5) - t}{2 {(t + 5)}^{\frac{3}{2}}}$ and $\sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}})$ .

$- \frac{(2 (t + 5) - t) \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}})}{2 {(t + 5)}^{\frac{3}{2}}}$

Step 4.28

Simplify.

Tap for more steps...

Step 4.28.1

Apply the distributive property.

$- \frac{(2 t + 2 \cdot 5 - t) \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}})}{2 {(t + 5)}^{\frac{3}{2}}}$

Step 4.28.2

Simplify the numerator.

Tap for more steps...

Step 4.28.2.1

Multiply $2$ by $5$ .

$- \frac{(2 t + 10 - t) \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}})}{2 {(t + 5)}^{\frac{3}{2}}}$

Step 4.28.2.2

Subtract $t$ from $2 t$ .

$- \frac{(t + 10) \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}})}{2 {(t + 5)}^{\frac{3}{2}}}$

Step 4.28.2.3

Apply the distributive property.

$- \frac{t \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) + 10 \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}})}{2 {(t + 5)}^{\frac{3}{2}}}$

Step 4.28.3

Factor $\sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}})$ out of $t \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) + 10 \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}})$ .

Tap for more steps...

Step 4.28.3.1

Factor $\sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}})$ out of $t \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}})$ .

$- \frac{\sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) t + 10 \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}})}{2 {(t + 5)}^{\frac{3}{2}}}$

Step 4.28.3.2

Factor $\sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}})$ out of $10 \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}})$ .

$- \frac{\sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) t + \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \cdot 10}{2 {(t + 5)}^{\frac{3}{2}}}$

Step 4.28.3.3

Factor $\sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}})$ out of $\sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) t + \sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) \cdot 10$ .

$- \frac{\sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) (t + 10)}{2 {(t + 5)}^{\frac{3}{2}}}$

Step 5

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

$q' = - \frac{\sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) (t + 10)}{2 {(t + 5)}^{\frac{3}{2}}}$

Step 6

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

$\frac{d q}{d t} = - \frac{\sin (\frac{t}{{(t + 5)}^{\frac{1}{2}}}) (t + 10)}{2 {(t + 5)}^{\frac{3}{2}}}$