Calculus Examples

$- 6 t {(5 t^{4} - 1)}^{4}$

Step 1

Since $- 6$ is constant with respect to $t$ , the derivative of $- 6 t {(5 t^{4} - 1)}^{4}$ with respect to $t$ is $- 6 \frac{d}{d t} [t {(5 t^{4} - 1)}^{4}]$ .

$- 6 \frac{d}{d t} [t {(5 t^{4} - 1)}^{4}]$

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 $g (t) = {(5 t^{4} - 1)}^{4}$ .

$- 6 (t \frac{d}{d t} [{(5 t^{4} - 1)}^{4}] + {(5 t^{4} - 1)}^{4} \frac{d}{d t} [t])$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{4}$ and $g (t) = 5 t^{4} - 1$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $5 t^{4} - 1$ .

$- 6 (t (\frac{d}{d u} [u^{4}] \frac{d}{d t} [5 t^{4} - 1]) + {(5 t^{4} - 1)}^{4} \frac{d}{d t} [t])$

Step 3.2

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

$- 6 (t (4 u^{3} \frac{d}{d t} [5 t^{4} - 1]) + {(5 t^{4} - 1)}^{4} \frac{d}{d t} [t])$

Step 3.3

Replace all occurrences of $u$ with $5 t^{4} - 1$ .

$- 6 (t (4 {(5 t^{4} - 1)}^{3} \frac{d}{d t} [5 t^{4} - 1]) + {(5 t^{4} - 1)}^{4} \frac{d}{d t} [t])$

Step 4

Differentiate.

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Step 4.1

By the Sum Rule, the derivative of $5 t^{4} - 1$ with respect to $t$ is $\frac{d}{d t} [5 t^{4}] + \frac{d}{d t} [- 1]$ .

$- 6 (t (4 {(5 t^{4} - 1)}^{3} (\frac{d}{d t} [5 t^{4}] + \frac{d}{d t} [- 1])) + {(5 t^{4} - 1)}^{4} \frac{d}{d t} [t])$

Step 4.2

Since $5$ is constant with respect to $t$ , the derivative of $5 t^{4}$ with respect to $t$ is $5 \frac{d}{d t} [t^{4}]$ .

$- 6 (t (4 {(5 t^{4} - 1)}^{3} (5 \frac{d}{d t} [t^{4}] + \frac{d}{d t} [- 1])) + {(5 t^{4} - 1)}^{4} \frac{d}{d t} [t])$

Step 4.3

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

$- 6 (t (4 {(5 t^{4} - 1)}^{3} (5 (4 t^{3}) + \frac{d}{d t} [- 1])) + {(5 t^{4} - 1)}^{4} \frac{d}{d t} [t])$

Step 4.4

Multiply $4$ by $5$ .

$- 6 (t (4 {(5 t^{4} - 1)}^{3} (20 t^{3} + \frac{d}{d t} [- 1])) + {(5 t^{4} - 1)}^{4} \frac{d}{d t} [t])$

Step 4.5

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

$- 6 (t (4 {(5 t^{4} - 1)}^{3} (20 t^{3} + 0)) + {(5 t^{4} - 1)}^{4} \frac{d}{d t} [t])$

Step 4.6

Simplify the expression.

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Step 4.6.1

Add $20 t^{3}$ and $0$ .

$- 6 (t (4 {(5 t^{4} - 1)}^{3} (20 t^{3})) + {(5 t^{4} - 1)}^{4} \frac{d}{d t} [t])$

Step 4.6.2

Multiply $20$ by $4$ .

$- 6 (t (80 {(5 t^{4} - 1)}^{3} t^{3}) + {(5 t^{4} - 1)}^{4} \frac{d}{d t} [t])$

Step 5

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

$- 6 (t^{3} t^{1} (80 {(5 t^{4} - 1)}^{3}) + {(5 t^{4} - 1)}^{4} \frac{d}{d t} [t])$

Step 6

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

$- 6 (t^{3 + 1} (80 {(5 t^{4} - 1)}^{3}) + {(5 t^{4} - 1)}^{4} \frac{d}{d t} [t])$

Step 7

Add $3$ and $1$ .

$- 6 (t^{4} (80 {(5 t^{4} - 1)}^{3}) + {(5 t^{4} - 1)}^{4} \frac{d}{d t} [t])$

Step 8

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

$- 6 (t^{4} (80 {(5 t^{4} - 1)}^{3}) + {(5 t^{4} - 1)}^{4} \cdot 1)$

Step 9

Multiply ${(5 t^{4} - 1)}^{4}$ by $1$ .

$- 6 (t^{4} (80 {(5 t^{4} - 1)}^{3}) + {(5 t^{4} - 1)}^{4})$

Step 10

Simplify.

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Step 10.1

Apply the distributive property.

$- 6 (t^{4} (80 {(5 t^{4} - 1)}^{3})) - 6 {(5 t^{4} - 1)}^{4}$

Step 10.2

Multiply $80$ by $- 6$ .

$- 480 (t^{4} ({(5 t^{4} - 1)}^{3})) - 6 {(5 t^{4} - 1)}^{4}$

Step 10.3

Factor $6 {(5 t^{4} - 1)}^{3}$ out of $- 480 t^{4} {(5 t^{4} - 1)}^{3} - 6 {(5 t^{4} - 1)}^{4}$ .

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Step 10.3.1

Factor $6 {(5 t^{4} - 1)}^{3}$ out of $- 480 t^{4} {(5 t^{4} - 1)}^{3}$ .

$6 {(5 t^{4} - 1)}^{3} (- 80 t^{4}) - 6 {(5 t^{4} - 1)}^{4}$

Step 10.3.2

Factor $6 {(5 t^{4} - 1)}^{3}$ out of $- 6 {(5 t^{4} - 1)}^{4}$ .

$6 {(5 t^{4} - 1)}^{3} (- 80 t^{4}) + 6 {(5 t^{4} - 1)}^{3} (- (5 t^{4} - 1))$

Step 10.3.3

Factor $6 {(5 t^{4} - 1)}^{3}$ out of $6 {(5 t^{4} - 1)}^{3} (- 80 t^{4}) + 6 {(5 t^{4} - 1)}^{3} (- (5 t^{4} - 1))$ .

$6 {(5 t^{4} - 1)}^{3} (- 80 t^{4} - (5 t^{4} - 1))$