Calculus Examples

$m (t) = - 7 t {(3 t^{4} - 1)}^{8}$

Step 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 $f (t) = - 7 t$ and $g (t) = {(3 t^{4} - 1)}^{8}$ .

$- 7 t \frac{d}{d t} [{(3 t^{4} - 1)}^{8}] + {(3 t^{4} - 1)}^{8} \frac{d}{d t} [- 7 t]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{8}$ and $g (t) = 3 t^{4} - 1$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $3 t^{4} - 1$ .

$- 7 t (\frac{d}{d u} [u^{8}] \frac{d}{d t} [3 t^{4} - 1]) + {(3 t^{4} - 1)}^{8} \frac{d}{d t} [- 7 t]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 8$ .

$- 7 t (8 u^{7} \frac{d}{d t} [3 t^{4} - 1]) + {(3 t^{4} - 1)}^{8} \frac{d}{d t} [- 7 t]$

Step 2.3

Replace all occurrences of $u$ with $3 t^{4} - 1$ .

$- 7 t (8 {(3 t^{4} - 1)}^{7} \frac{d}{d t} [3 t^{4} - 1]) + {(3 t^{4} - 1)}^{8} \frac{d}{d t} [- 7 t]$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $3 t^{4} - 1$ with respect to $t$ is $\frac{d}{d t} [3 t^{4}] + \frac{d}{d t} [- 1]$ .

$- 7 t (8 {(3 t^{4} - 1)}^{7} (\frac{d}{d t} [3 t^{4}] + \frac{d}{d t} [- 1])) + {(3 t^{4} - 1)}^{8} \frac{d}{d t} [- 7 t]$

Step 3.2

Since $3$ is constant with respect to $t$ , the derivative of $3 t^{4}$ with respect to $t$ is $3 \frac{d}{d t} [t^{4}]$ .

$- 7 t (8 {(3 t^{4} - 1)}^{7} (3 \frac{d}{d t} [t^{4}] + \frac{d}{d t} [- 1])) + {(3 t^{4} - 1)}^{8} \frac{d}{d t} [- 7 t]$

Step 3.3

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 4$ .

$- 7 t (8 {(3 t^{4} - 1)}^{7} (3 (4 t^{3}) + \frac{d}{d t} [- 1])) + {(3 t^{4} - 1)}^{8} \frac{d}{d t} [- 7 t]$

Step 3.4

Multiply $4$ by $3$ .

$- 7 t (8 {(3 t^{4} - 1)}^{7} (12 t^{3} + \frac{d}{d t} [- 1])) + {(3 t^{4} - 1)}^{8} \frac{d}{d t} [- 7 t]$

Step 3.5

Since $- 1$ is constant with respect to $t$ , the derivative of $- 1$ with respect to $t$ is $0$ .

$- 7 t (8 {(3 t^{4} - 1)}^{7} (12 t^{3} + 0)) + {(3 t^{4} - 1)}^{8} \frac{d}{d t} [- 7 t]$

Step 3.6

Simplify the expression.

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Step 3.6.1

Add $12 t^{3}$ and $0$ .

$- 7 t (8 {(3 t^{4} - 1)}^{7} (12 t^{3})) + {(3 t^{4} - 1)}^{8} \frac{d}{d t} [- 7 t]$

Step 3.6.2

Multiply $12$ by $8$ .

$- 7 t (96 {(3 t^{4} - 1)}^{7} t^{3}) + {(3 t^{4} - 1)}^{8} \frac{d}{d t} [- 7 t]$

Step 3.6.3

Reorder the factors of $96 {(3 t^{4} - 1)}^{7} t^{3}$ .

$- 7 t (96 t^{3} {(3 t^{4} - 1)}^{7}) + {(3 t^{4} - 1)}^{8} \frac{d}{d t} [- 7 t]$

Step 3.7

Since $- 7$ is constant with respect to $t$ , the derivative of $- 7 t$ with respect to $t$ is $- 7 \frac{d}{d t} [t]$ .

$- 7 t (96 t^{3} {(3 t^{4} - 1)}^{7}) + {(3 t^{4} - 1)}^{8} (- 7 \frac{d}{d t} [t])$

Step 3.8

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

$- 7 t (96 t^{3} {(3 t^{4} - 1)}^{7}) + {(3 t^{4} - 1)}^{8} (- 7 \cdot 1)$

Step 3.9

Multiply $- 7$ by $1$ .

$- 7 t (96 t^{3} {(3 t^{4} - 1)}^{7}) + {(3 t^{4} - 1)}^{8} \cdot - 7$

Step 4

Simplify.

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Step 4.1

Multiply $96$ by $- 7$ .

$- 672 t (t^{3} {(3 t^{4} - 1)}^{7}) + {(3 t^{4} - 1)}^{8} \cdot - 7$

Step 4.2

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

$- 672 (t^{3} t^{1}) {(3 t^{4} - 1)}^{7} + {(3 t^{4} - 1)}^{8} \cdot - 7$

Step 4.3

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

$- 672 t^{3 + 1} {(3 t^{4} - 1)}^{7} + {(3 t^{4} - 1)}^{8} \cdot - 7$

Step 4.4

Add $3$ and $1$ .

$- 672 t^{4} {(3 t^{4} - 1)}^{7} + {(3 t^{4} - 1)}^{8} \cdot - 7$

Step 4.5

Move $- 7$ to the left of ${(3 t^{4} - 1)}^{8}$ .

$- 672 t^{4} {(3 t^{4} - 1)}^{7} - 7 \cdot {(3 t^{4} - 1)}^{8}$

Step 4.6

Factor $7 {(3 t^{4} - 1)}^{7}$ out of $- 672 t^{4} {(3 t^{4} - 1)}^{7} - 7 {(3 t^{4} - 1)}^{8}$ .

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Step 4.6.1

Factor $7 {(3 t^{4} - 1)}^{7}$ out of $- 672 t^{4} {(3 t^{4} - 1)}^{7}$ .

$7 {(3 t^{4} - 1)}^{7} (- 96 t^{4}) - 7 {(3 t^{4} - 1)}^{8}$

Step 4.6.2

Factor $7 {(3 t^{4} - 1)}^{7}$ out of $- 7 {(3 t^{4} - 1)}^{8}$ .

$7 {(3 t^{4} - 1)}^{7} (- 96 t^{4}) + 7 {(3 t^{4} - 1)}^{7} (- (3 t^{4} - 1))$

Step 4.6.3

Factor $7 {(3 t^{4} - 1)}^{7}$ out of $7 {(3 t^{4} - 1)}^{7} (- 96 t^{4}) + 7 {(3 t^{4} - 1)}^{7} (- (3 t^{4} - 1))$ .

$7 {(3 t^{4} - 1)}^{7} (- 96 t^{4} - (3 t^{4} - 1))$