Calculus Examples

$2 {(t^{2} + 11 t)}^{- 7}$

Step 1

Since $2$ is constant with respect to $t$ , the derivative of $2 {(t^{2} + 11 t)}^{- 7}$ with respect to $t$ is $2 \frac{d}{d t} [{(t^{2} + 11 t)}^{- 7}]$ .

$2 \frac{d}{d t} [{(t^{2} + 11 t)}^{- 7}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{- 7}$ and $g (t) = t^{2} + 11 t$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $t^{2} + 11 t$ .

$2 (\frac{d}{d u} [u^{- 7}] \frac{d}{d t} [t^{2} + 11 t])$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = - 7$ .

$2 (- 7 u^{- 8} \frac{d}{d t} [t^{2} + 11 t])$

Step 2.3

Replace all occurrences of $u$ with $t^{2} + 11 t$ .

$2 (- 7 {(t^{2} + 11 t)}^{- 8} \frac{d}{d t} [t^{2} + 11 t])$

Step 3

Differentiate.

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Step 3.1

Multiply $- 7$ by $2$ .

$- 14 ({(t^{2} + 11 t)}^{- 8} \frac{d}{d t} [t^{2} + 11 t])$

Step 3.2

By the Sum Rule, the derivative of $t^{2} + 11 t$ with respect to $t$ is $\frac{d}{d t} [t^{2}] + \frac{d}{d t} [11 t]$ .

$- 14 {(t^{2} + 11 t)}^{- 8} (\frac{d}{d t} [t^{2}] + \frac{d}{d t} [11 t])$

Step 3.3

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

$- 14 {(t^{2} + 11 t)}^{- 8} (2 t + \frac{d}{d t} [11 t])$

Step 3.4

Since $11$ is constant with respect to $t$ , the derivative of $11 t$ with respect to $t$ is $11 \frac{d}{d t} [t]$ .

$- 14 {(t^{2} + 11 t)}^{- 8} (2 t + 11 \frac{d}{d t} [t])$

Step 3.5

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

$- 14 {(t^{2} + 11 t)}^{- 8} (2 t + 11 \cdot 1)$

Step 3.6

Multiply $11$ by $1$ .

$- 14 {(t^{2} + 11 t)}^{- 8} (2 t + 11)$

Step 4

Simplify.

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Step 4.1

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

$- 14 \frac{1}{{(t^{2} + 11 t)}^{8}} (2 t + 11)$

Step 4.2

Combine terms.

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Step 4.2.1

Combine $- 14$ and $\frac{1}{{(t^{2} + 11 t)}^{8}}$ .

$\frac{- 14}{{(t^{2} + 11 t)}^{8}} (2 t + 11)$

Step 4.2.2

Move the negative in front of the fraction.

$- \frac{14}{{(t^{2} + 11 t)}^{8}} (2 t + 11)$

Step 4.3

Reorder the factors of $- \frac{14}{{(t^{2} + 11 t)}^{8}} (2 t + 11)$ .

$- (2 t + 11) \frac{14}{{(t^{2} + 11 t)}^{8}}$

Step 4.4

Apply the distributive property.

$(- (2 t) - 1 \cdot 11) \frac{14}{{(t^{2} + 11 t)}^{8}}$

Step 4.5

Multiply $2$ by $- 1$ .

$(- 2 t - 1 \cdot 11) \frac{14}{{(t^{2} + 11 t)}^{8}}$

Step 4.6

Multiply $- 1$ by $11$ .

$(- 2 t - 11) \frac{14}{{(t^{2} + 11 t)}^{8}}$

Step 4.7

Simplify the denominator.

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Step 4.7.1

Factor $t$ out of $t^{2} + 11 t$ .

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Step 4.7.1.1

Factor $t$ out of $t^{2}$ .

$(- 2 t - 11) \frac{14}{{(t \cdot t + 11 t)}^{8}}$

Step 4.7.1.2

Factor $t$ out of $11 t$ .

$(- 2 t - 11) \frac{14}{{(t \cdot t + t \cdot 11)}^{8}}$

Step 4.7.1.3

Factor $t$ out of $t \cdot t + t \cdot 11$ .

$(- 2 t - 11) \frac{14}{{(t (t + 11))}^{8}}$

Step 4.7.2

Apply the product rule to $t (t + 11)$ .

$(- 2 t - 11) \frac{14}{t^{8} {(t + 11)}^{8}}$

Step 4.8

Multiply $- 2 t - 11$ by $\frac{14}{t^{8} {(t + 11)}^{8}}$ .

$\frac{(- 2 t - 11) \cdot 14}{t^{8} {(t + 11)}^{8}}$

Step 4.9

Move $14$ to the left of $- 2 t - 11$ .

$\frac{14 \cdot (- 2 t - 11)}{t^{8} {(t + 11)}^{8}}$

Step 4.10

Factor $- 1$ out of $- 2 t$ .

$\frac{14 (- (2 t) - 11)}{t^{8} {(t + 11)}^{8}}$

Step 4.11

Rewrite $- 11$ as $- 1 (11)$ .

$\frac{14 (- (2 t) - 1 (11))}{t^{8} {(t + 11)}^{8}}$

Step 4.12

Factor $- 1$ out of $- (2 t) - 1 (11)$ .

$\frac{14 (- (2 t + 11))}{t^{8} {(t + 11)}^{8}}$

Step 4.13

Rewrite $- (2 t + 11)$ as $- 1 (2 t + 11)$ .

$\frac{14 (- 1 (2 t + 11))}{t^{8} {(t + 11)}^{8}}$

Step 4.14

Move the negative in front of the fraction.

$- \frac{14 (2 t + 11)}{t^{8} {(t + 11)}^{8}}$