Calculus Examples

$w = {(t^{7} + 1)}^{90}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d t} (w) = \frac{d}{d t} ({(t^{7} + 1)}^{90})$

Step 2

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

$w'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{90}$ and $g (t) = t^{7} + 1$ .

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Step 3.1.1

To apply the Chain Rule, set $u$ as $t^{7} + 1$ .

$\frac{d}{d u} [u^{90}] \frac{d}{d t} [t^{7} + 1]$

Step 3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 90$ .

$90 u^{89} \frac{d}{d t} [t^{7} + 1]$

Step 3.1.3

Replace all occurrences of $u$ with $t^{7} + 1$ .

$90 {(t^{7} + 1)}^{89} \frac{d}{d t} [t^{7} + 1]$

Step 3.2

Differentiate.

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Step 3.2.1

By the Sum Rule, the derivative of $t^{7} + 1$ with respect to $t$ is $\frac{d}{d t} [t^{7}] + \frac{d}{d t} [1]$ .

$90 {(t^{7} + 1)}^{89} (\frac{d}{d t} [t^{7}] + \frac{d}{d t} [1])$

Step 3.2.2

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

$90 {(t^{7} + 1)}^{89} (7 t^{6} + \frac{d}{d t} [1])$

Step 3.2.3

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

$90 {(t^{7} + 1)}^{89} (7 t^{6} + 0)$

Step 3.2.4

Simplify the expression.

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Step 3.2.4.1

Add $7 t^{6}$ and $0$ .

$90 {(t^{7} + 1)}^{89} (7 t^{6})$

Step 3.2.4.2

Multiply $7$ by $90$ .

$630 {(t^{7} + 1)}^{89} t^{6}$

Step 3.2.4.3

Reorder the factors of $630 {(t^{7} + 1)}^{89} t^{6}$ .

$630 t^{6} {(t^{7} + 1)}^{89}$

Step 4

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

$w' = 630 t^{6} {(t^{7} + 1)}^{89}$

Step 5

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

$\frac{d w}{d t} = 630 t^{6} {(t^{7} + 1)}^{89}$