Calculus Examples

$z = (4 - t) {(9 + t^{2})}^{- 1}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d t} (z) = \frac{d}{d t} ((4 - t) {(9 + t^{2})}^{- 1})$

Step 2

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

$z'$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

Differentiate using the Product Rule which states that $\frac{d}{d t} [f (t) g (t)]$ is $f (t) \frac{d}{d t} [g (t)] + g (t) \frac{d}{d t} [f (t)]$ where $f (t) = 4 - t$ and $g (t) = {(9 + t^{2})}^{- 1}$ .

$(4 - t) \frac{d}{d t} [{(9 + t^{2})}^{- 1}] + {(9 + t^{2})}^{- 1} \frac{d}{d t} [4 - t]$

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{- 1}$ and $g (t) = 9 + t^{2}$ .

Tap for more steps...

Step 3.2.1

To apply the Chain Rule, set $u$ as $9 + t^{2}$ .

$(4 - t) (\frac{d}{d u} [u^{- 1}] \frac{d}{d t} [9 + t^{2}]) + {(9 + t^{2})}^{- 1} \frac{d}{d t} [4 - t]$

Step 3.2.2

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

$(4 - t) (- u^{- 2} \frac{d}{d t} [9 + t^{2}]) + {(9 + t^{2})}^{- 1} \frac{d}{d t} [4 - t]$

Step 3.2.3

Replace all occurrences of $u$ with $9 + t^{2}$ .

$(4 - t) (- {(9 + t^{2})}^{- 2} \frac{d}{d t} [9 + t^{2}]) + {(9 + t^{2})}^{- 1} \frac{d}{d t} [4 - t]$

Step 3.3

Differentiate.

Tap for more steps...

Step 3.3.1

By the Sum Rule, the derivative of $9 + t^{2}$ with respect to $t$ is $\frac{d}{d t} [9] + \frac{d}{d t} [t^{2}]$ .

$(4 - t) (- {(9 + t^{2})}^{- 2} (\frac{d}{d t} [9] + \frac{d}{d t} [t^{2}])) + {(9 + t^{2})}^{- 1} \frac{d}{d t} [4 - t]$

Step 3.3.2

Since $9$ is constant with respect to $t$ , the derivative of $9$ with respect to $t$ is $0$ .

$(4 - t) (- {(9 + t^{2})}^{- 2} (0 + \frac{d}{d t} [t^{2}])) + {(9 + t^{2})}^{- 1} \frac{d}{d t} [4 - t]$

Step 3.3.3

Add $0$ and $\frac{d}{d t} [t^{2}]$ .

$(4 - t) (- {(9 + t^{2})}^{- 2} \frac{d}{d t} [t^{2}]) + {(9 + t^{2})}^{- 1} \frac{d}{d t} [4 - t]$

Step 3.3.4

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

$(4 - t) (- {(9 + t^{2})}^{- 2} (2 t)) + {(9 + t^{2})}^{- 1} \frac{d}{d t} [4 - t]$

Step 3.3.5

Multiply $2$ by $- 1$ .

$(4 - t) (- 2 {(9 + t^{2})}^{- 2} t) + {(9 + t^{2})}^{- 1} \frac{d}{d t} [4 - t]$

Step 3.3.6

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

$(4 - t) (- 2 {(9 + t^{2})}^{- 2} t) + {(9 + t^{2})}^{- 1} (\frac{d}{d t} [4] + \frac{d}{d t} [- t])$

Step 3.3.7

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

$(4 - t) (- 2 {(9 + t^{2})}^{- 2} t) + {(9 + t^{2})}^{- 1} (0 + \frac{d}{d t} [- t])$

Step 3.3.8

Add $0$ and $\frac{d}{d t} [- t]$ .

$(4 - t) (- 2 {(9 + t^{2})}^{- 2} t) + {(9 + t^{2})}^{- 1} \frac{d}{d t} [- t]$

Step 3.3.9

Since $- 1$ is constant with respect to $t$ , the derivative of $- t$ with respect to $t$ is $- \frac{d}{d t} [t]$ .

$(4 - t) (- 2 {(9 + t^{2})}^{- 2} t) + {(9 + t^{2})}^{- 1} (- \frac{d}{d t} [t])$

Step 3.3.10

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

$(4 - t) (- 2 {(9 + t^{2})}^{- 2} t) + {(9 + t^{2})}^{- 1} (- 1 \cdot 1)$

Step 3.3.11

Simplify the expression.

Tap for more steps...

Step 3.3.11.1

Multiply $- 1$ by $1$ .

$(4 - t) (- 2 {(9 + t^{2})}^{- 2} t) + {(9 + t^{2})}^{- 1} \cdot - 1$

Step 3.3.11.2

Move $- 1$ to the left of ${(9 + t^{2})}^{- 1}$ .

$(4 - t) (- 2 {(9 + t^{2})}^{- 2} t) - 1 \cdot {(9 + t^{2})}^{- 1}$

Step 3.3.11.3

Rewrite $- 1 {(9 + t^{2})}^{- 1}$ as $- {(9 + t^{2})}^{- 1}$ .

$(4 - t) (- 2 {(9 + t^{2})}^{- 2} t) - {(9 + t^{2})}^{- 1}$

Step 3.4

Simplify.

Tap for more steps...

Step 3.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$(4 - t) (- 2 \frac{1}{{(9 + t^{2})}^{2}} t) - {(9 + t^{2})}^{- 1}$

Step 3.4.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$(4 - t) (- 2 \frac{1}{{(9 + t^{2})}^{2}} t) - \frac{1}{9 + t^{2}}$

Step 3.4.3

Combine terms.

Tap for more steps...

Step 3.4.3.1

Combine $- 2$ and $\frac{1}{{(9 + t^{2})}^{2}}$ .

$(4 - t) (\frac{- 2}{{(9 + t^{2})}^{2}} t) - \frac{1}{9 + t^{2}}$

Step 3.4.3.2

Move the negative in front of the fraction.

$(4 - t) (- \frac{2}{{(9 + t^{2})}^{2}} t) - \frac{1}{9 + t^{2}}$

Step 3.4.3.3

Combine $t$ and $\frac{2}{{(9 + t^{2})}^{2}}$ .

$(4 - t) (- \frac{t \cdot 2}{{(9 + t^{2})}^{2}}) - \frac{1}{9 + t^{2}}$

Step 3.4.3.4

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

$(4 - t) (- \frac{2 \cdot t}{{(9 + t^{2})}^{2}}) - \frac{1}{9 + t^{2}}$

Step 3.4.3.5

To write $(4 - t) (- \frac{2 t}{{(9 + t^{2})}^{2}})$ as a fraction with a common denominator, multiply by $\frac{9 + t^{2}}{9 + t^{2}}$ .

$(4 - t) (- \frac{2 t}{{(9 + t^{2})}^{2}}) \cdot \frac{9 + t^{2}}{9 + t^{2}} - \frac{1}{9 + t^{2}}$

Step 3.4.3.6

Combine $(4 - t) (- \frac{2 t}{{(9 + t^{2})}^{2}})$ and $\frac{9 + t^{2}}{9 + t^{2}}$ .

$\frac{(4 - t) (- \frac{2 t}{{(9 + t^{2})}^{2}}) (9 + t^{2})}{9 + t^{2}} - \frac{1}{9 + t^{2}}$

Step 3.4.3.7

Combine the numerators over the common denominator.

$\frac{(4 - t) (- \frac{2 t}{{(9 + t^{2})}^{2}}) (9 + t^{2}) - 1}{9 + t^{2}}$

Step 3.4.4

Reorder terms.

$\frac{- (4 - t) (9 + t^{2}) \frac{2 t}{{(9 + t^{2})}^{2}} - 1}{t^{2} + 9}$

Step 3.4.5

Simplify the numerator.

Tap for more steps...

Step 3.4.5.1

Factor $- 1$ out of $- (4 - t) (9 + t^{2}) \frac{2 t}{{(9 + t^{2})}^{2}} - 1$ .

Tap for more steps...

Step 3.4.5.1.1

Reorder the expression.

Tap for more steps...

Step 3.4.5.1.1.1

Reorder $4$ and $- t$ .

$\frac{- (- t + 4) (9 + t^{2}) \frac{2 t}{{(9 + t^{2})}^{2}} - 1}{t^{2} + 9}$

Step 3.4.5.1.1.2

Reorder $9$ and $t^{2}$ .

$\frac{- (- t + 4) (t^{2} + 9) \frac{2 t}{{(9 + t^{2})}^{2}} - 1}{t^{2} + 9}$

Step 3.4.5.1.1.3

Reorder $9$ and $t^{2}$ .

$\frac{- (- t + 4) (t^{2} + 9) \frac{2 t}{{(t^{2} + 9)}^{2}} - 1}{t^{2} + 9}$

Step 3.4.5.1.2

Factor $- 1$ out of $- (- t + 4) (t^{2} + 9) \frac{2 t}{{(t^{2} + 9)}^{2}}$ .

$\frac{- (((- t + 4) (t^{2} + 9)) \frac{2 t}{{(t^{2} + 9)}^{2}}) - 1}{t^{2} + 9}$

Step 3.4.5.1.3

Rewrite $- 1$ as $- 1 (1)$ .

$\frac{- (((- t + 4) (t^{2} + 9)) \frac{2 t}{{(t^{2} + 9)}^{2}}) - 1 (1)}{t^{2} + 9}$

Step 3.4.5.1.4

Factor $- 1$ out of $- (((- t + 4) (t^{2} + 9)) \frac{2 t}{{(t^{2} + 9)}^{2}}) - 1 (1)$ .

$\frac{- (((- t + 4) (t^{2} + 9)) \frac{2 t}{{(t^{2} + 9)}^{2}} + 1)}{t^{2} + 9}$

Step 3.4.5.2

Cancel the common factor of $t^{2} + 9$ .

Tap for more steps...

Step 3.4.5.2.1

Factor $t^{2} + 9$ out of $(- t + 4) (t^{2} + 9)$ .

$\frac{- ((t^{2} + 9) (- t + 4) \frac{2 t}{{(t^{2} + 9)}^{2}} + 1)}{t^{2} + 9}$

Step 3.4.5.2.2

Factor $t^{2} + 9$ out of ${(t^{2} + 9)}^{2}$ .

$\frac{- ((t^{2} + 9) (- t + 4) \frac{2 t}{(t^{2} + 9) (t^{2} + 9)} + 1)}{t^{2} + 9}$

Step 3.4.5.2.3

Cancel the common factor.

$\frac{- ((t^{2} + 9) (- t + 4) \frac{2 t}{(t^{2} + 9) (t^{2} + 9)} + 1)}{t^{2} + 9}$

Step 3.4.5.2.4

Rewrite the expression.

$\frac{- ((- t + 4) \frac{2 t}{t^{2} + 9} + 1)}{t^{2} + 9}$

Step 3.4.5.3

Multiply $- t + 4$ by $\frac{2 t}{t^{2} + 9}$ .

$\frac{- (\frac{(- t + 4) (2 t)}{t^{2} + 9} + 1)}{t^{2} + 9}$

Step 3.4.5.4

Move $2$ to the left of $- t + 4$ .

$\frac{- (\frac{2 \cdot (- t + 4) t}{t^{2} + 9} + 1)}{t^{2} + 9}$

Step 3.4.5.5

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

$\frac{- (\frac{2 (- t + 4) t}{t^{2} + 9} + \frac{t^{2} + 9}{t^{2} + 9})}{t^{2} + 9}$

Step 3.4.5.6

Combine the numerators over the common denominator.

$\frac{- \frac{2 (- t + 4) t + t^{2} + 9}{t^{2} + 9}}{t^{2} + 9}$

Step 3.4.5.7

Reorder terms.

$\frac{- \frac{t^{2} + 2 t (- t + 4) + 9}{t^{2} + 9}}{t^{2} + 9}$

Step 3.4.5.8

Rewrite $\frac{t^{2} + 2 t (- t + 4) + 9}{t^{2} + 9}$ in a factored form.

Tap for more steps...

Step 3.4.5.8.1

Apply the distributive property.

$\frac{- \frac{t^{2} + 2 t (- t) + 2 t \cdot 4 + 9}{t^{2} + 9}}{t^{2} + 9}$

Step 3.4.5.8.2

Rewrite using the commutative property of multiplication.

$\frac{- \frac{t^{2} + 2 \cdot - 1 t \cdot t + 2 t \cdot 4 + 9}{t^{2} + 9}}{t^{2} + 9}$

Step 3.4.5.8.3

Multiply $4$ by $2$ .

$\frac{- \frac{t^{2} + 2 \cdot - 1 t \cdot t + 8 t + 9}{t^{2} + 9}}{t^{2} + 9}$

Step 3.4.5.8.4

Simplify each term.

Tap for more steps...

Step 3.4.5.8.4.1

Multiply $t$ by $t$ by adding the exponents.

Tap for more steps...

Step 3.4.5.8.4.1.1

Move $t$ .

$\frac{- \frac{t^{2} + 2 \cdot - 1 (t \cdot t) + 8 t + 9}{t^{2} + 9}}{t^{2} + 9}$

Step 3.4.5.8.4.1.2

Multiply $t$ by $t$ .

$\frac{- \frac{t^{2} + 2 \cdot - 1 t^{2} + 8 t + 9}{t^{2} + 9}}{t^{2} + 9}$

Step 3.4.5.8.4.2

Multiply $2$ by $- 1$ .

$\frac{- \frac{t^{2} - 2 t^{2} + 8 t + 9}{t^{2} + 9}}{t^{2} + 9}$

Step 3.4.5.8.5

Subtract $2 t^{2}$ from $t^{2}$ .

$\frac{- \frac{- t^{2} + 8 t + 9}{t^{2} + 9}}{t^{2} + 9}$

Step 3.4.5.8.6

Factor by grouping.

Tap for more steps...

Step 3.4.5.8.6.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 9 = - 9$ and whose sum is $b = 8$ .

Tap for more steps...

Step 3.4.5.8.6.1.1

Factor $8$ out of $8 t$ .

$\frac{- \frac{- t^{2} + 8 (t) + 9}{t^{2} + 9}}{t^{2} + 9}$

Step 3.4.5.8.6.1.2

Rewrite $8$ as $- 1$ plus $9$

$\frac{- \frac{- t^{2} + (- 1 + 9) t + 9}{t^{2} + 9}}{t^{2} + 9}$

Step 3.4.5.8.6.1.3

Apply the distributive property.

$\frac{- \frac{- t^{2} - 1 t + 9 t + 9}{t^{2} + 9}}{t^{2} + 9}$

Step 3.4.5.8.6.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 3.4.5.8.6.2.1

Group the first two terms and the last two terms.

$\frac{- \frac{(- t^{2} - 1 t) + 9 t + 9}{t^{2} + 9}}{t^{2} + 9}$

Step 3.4.5.8.6.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{- \frac{t (- t - 1) - 9 (- t - 1)}{t^{2} + 9}}{t^{2} + 9}$

Step 3.4.5.8.6.3

Factor the polynomial by factoring out the greatest common factor, $- t - 1$ .

$\frac{- \frac{(- t - 1) (t - 9)}{t^{2} + 9}}{t^{2} + 9}$

Step 3.4.6

Multiply the numerator by the reciprocal of the denominator.

$- \frac{(- t - 1) (t - 9)}{t^{2} + 9} \cdot \frac{1}{t^{2} + 9}$

Step 3.4.7

Multiply $- \frac{(- t - 1) (t - 9)}{t^{2} + 9} \cdot \frac{1}{t^{2} + 9}$ .

Tap for more steps...

Step 3.4.7.1

Multiply $\frac{1}{t^{2} + 9}$ by $\frac{(- t - 1) (t - 9)}{t^{2} + 9}$ .

$- \frac{(- t - 1) (t - 9)}{(t^{2} + 9) (t^{2} + 9)}$

Step 3.4.7.2

Raise $t^{2} + 9$ to the power of $1$ .

$- \frac{(- t - 1) (t - 9)}{{(t^{2} + 9)}^{1} (t^{2} + 9)}$

Step 3.4.7.3

Raise $t^{2} + 9$ to the power of $1$ .

$- \frac{(- t - 1) (t - 9)}{{(t^{2} + 9)}^{1} {(t^{2} + 9)}^{1}}$

Step 3.4.7.4

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

$- \frac{(- t - 1) (t - 9)}{{(t^{2} + 9)}^{1 + 1}}$

Step 3.4.7.5

Add $1$ and $1$ .

$- \frac{(- t - 1) (t - 9)}{{(t^{2} + 9)}^{2}}$

Step 3.4.8

Factor $- 1$ out of $- t$ .

$- \frac{(- (t) - 1) (t - 9)}{{(t^{2} + 9)}^{2}}$

Step 3.4.9

Rewrite $- 1$ as $- 1 (1)$ .

$- \frac{(- (t) - 1 (1)) (t - 9)}{{(t^{2} + 9)}^{2}}$

Step 3.4.10

Factor $- 1$ out of $- (t) - 1 (1)$ .

$- \frac{- (t + 1) (t - 9)}{{(t^{2} + 9)}^{2}}$

Step 3.4.11

Rewrite $- (t + 1)$ as $- 1 (t + 1)$ .

$- \frac{- 1 (t + 1) (t - 9)}{{(t^{2} + 9)}^{2}}$

Step 3.4.12

Move the negative in front of the fraction.

$- - \frac{(t + 1) (t - 9)}{{(t^{2} + 9)}^{2}}$

Step 3.4.13

Multiply $- 1$ by $- 1$ .

$1 \frac{(t + 1) (t - 9)}{{(t^{2} + 9)}^{2}}$

Step 3.4.14

Multiply $\frac{(t + 1) (t - 9)}{{(t^{2} + 9)}^{2}}$ by $1$ .

$\frac{(t + 1) (t - 9)}{{(t^{2} + 9)}^{2}}$

Step 4

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

$z' = \frac{(t + 1) (t - 9)}{{(t^{2} + 9)}^{2}}$

Step 5

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

$\frac{d z}{d t} = \frac{(t + 1) (t - 9)}{{(t^{2} + 9)}^{2}}$