Calculus Examples

$f (t) = - 3 t {(5 t^{8} - 1)}^{7}$

Step 1

Since $- 3$ is constant with respect to $t$ , the derivative of $- 3 t {(5 t^{8} - 1)}^{7}$ with respect to $t$ is $- 3 \frac{d}{d t} [t {(5 t^{8} - 1)}^{7}]$ .

$- 3 \frac{d}{d t} [t {(5 t^{8} - 1)}^{7}]$

Step 2

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 $f (t) = t$ and $g (t) = {(5 t^{8} - 1)}^{7}$ .

$- 3 (t \frac{d}{d t} [{(5 t^{8} - 1)}^{7}] + {(5 t^{8} - 1)}^{7} \frac{d}{d t} [t])$

Step 3

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 $g (t) = 5 t^{8} - 1$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $5 t^{8} - 1$ .

$- 3 (t (\frac{d}{d u} [u^{7}] \frac{d}{d t} [5 t^{8} - 1]) + {(5 t^{8} - 1)}^{7} \frac{d}{d t} [t])$

Step 3.2

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

$- 3 (t (7 u^{6} \frac{d}{d t} [5 t^{8} - 1]) + {(5 t^{8} - 1)}^{7} \frac{d}{d t} [t])$

Step 3.3

Replace all occurrences of $u$ with $5 t^{8} - 1$ .

$- 3 (t (7 {(5 t^{8} - 1)}^{6} \frac{d}{d t} [5 t^{8} - 1]) + {(5 t^{8} - 1)}^{7} \frac{d}{d t} [t])$

Step 4

Differentiate.

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Step 4.1

By the Sum Rule, the derivative of $5 t^{8} - 1$ with respect to $t$ is $\frac{d}{d t} [5 t^{8}] + \frac{d}{d t} [- 1]$ .

$- 3 (t (7 {(5 t^{8} - 1)}^{6} (\frac{d}{d t} [5 t^{8}] + \frac{d}{d t} [- 1])) + {(5 t^{8} - 1)}^{7} \frac{d}{d t} [t])$

Step 4.2

Since $5$ is constant with respect to $t$ , the derivative of $5 t^{8}$ with respect to $t$ is $5 \frac{d}{d t} [t^{8}]$ .

$- 3 (t (7 {(5 t^{8} - 1)}^{6} (5 \frac{d}{d t} [t^{8}] + \frac{d}{d t} [- 1])) + {(5 t^{8} - 1)}^{7} \frac{d}{d t} [t])$

Step 4.3

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

$- 3 (t (7 {(5 t^{8} - 1)}^{6} (5 (8 t^{7}) + \frac{d}{d t} [- 1])) + {(5 t^{8} - 1)}^{7} \frac{d}{d t} [t])$

Step 4.4

Multiply $8$ by $5$ .

$- 3 (t (7 {(5 t^{8} - 1)}^{6} (40 t^{7} + \frac{d}{d t} [- 1])) + {(5 t^{8} - 1)}^{7} \frac{d}{d t} [t])$

Step 4.5

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

$- 3 (t (7 {(5 t^{8} - 1)}^{6} (40 t^{7} + 0)) + {(5 t^{8} - 1)}^{7} \frac{d}{d t} [t])$

Step 4.6

Simplify the expression.

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Step 4.6.1

Add $40 t^{7}$ and $0$ .

$- 3 (t (7 {(5 t^{8} - 1)}^{6} (40 t^{7})) + {(5 t^{8} - 1)}^{7} \frac{d}{d t} [t])$

Step 4.6.2

Multiply $40$ by $7$ .

$- 3 (t (280 {(5 t^{8} - 1)}^{6} t^{7}) + {(5 t^{8} - 1)}^{7} \frac{d}{d t} [t])$

Step 5

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

$- 3 (t^{7} t^{1} (280 {(5 t^{8} - 1)}^{6}) + {(5 t^{8} - 1)}^{7} \frac{d}{d t} [t])$

Step 6

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

$- 3 (t^{7 + 1} (280 {(5 t^{8} - 1)}^{6}) + {(5 t^{8} - 1)}^{7} \frac{d}{d t} [t])$

Step 7

Add $7$ and $1$ .

$- 3 (t^{8} (280 {(5 t^{8} - 1)}^{6}) + {(5 t^{8} - 1)}^{7} \frac{d}{d t} [t])$

Step 8

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

$- 3 (t^{8} (280 {(5 t^{8} - 1)}^{6}) + {(5 t^{8} - 1)}^{7} \cdot 1)$

Step 9

Multiply ${(5 t^{8} - 1)}^{7}$ by $1$ .

$- 3 (t^{8} (280 {(5 t^{8} - 1)}^{6}) + {(5 t^{8} - 1)}^{7})$

Step 10

Simplify.

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Step 10.1

Apply the distributive property.

$- 3 (t^{8} (280 {(5 t^{8} - 1)}^{6})) - 3 {(5 t^{8} - 1)}^{7}$

Step 10.2

Multiply $280$ by $- 3$ .

$- 840 (t^{8} ({(5 t^{8} - 1)}^{6})) - 3 {(5 t^{8} - 1)}^{7}$

Step 10.3

Factor $3 {(5 t^{8} - 1)}^{6}$ out of $- 840 t^{8} {(5 t^{8} - 1)}^{6} - 3 {(5 t^{8} - 1)}^{7}$ .

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Step 10.3.1

Factor $3 {(5 t^{8} - 1)}^{6}$ out of $- 840 t^{8} {(5 t^{8} - 1)}^{6}$ .

$3 {(5 t^{8} - 1)}^{6} (- 280 t^{8}) - 3 {(5 t^{8} - 1)}^{7}$

Step 10.3.2

Factor $3 {(5 t^{8} - 1)}^{6}$ out of $- 3 {(5 t^{8} - 1)}^{7}$ .

$3 {(5 t^{8} - 1)}^{6} (- 280 t^{8}) + 3 {(5 t^{8} - 1)}^{6} (- (5 t^{8} - 1))$

Step 10.3.3

Factor $3 {(5 t^{8} - 1)}^{6}$ out of $3 {(5 t^{8} - 1)}^{6} (- 280 t^{8}) + 3 {(5 t^{8} - 1)}^{6} (- (5 t^{8} - 1))$ .

$3 {(5 t^{8} - 1)}^{6} (- 280 t^{8} - (5 t^{8} - 1))$