Calculus Examples

$y = {(x^{4})}^{10}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{10}$ and $g (x) = x^{4}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $x^{4}$ .

$\frac{d}{d u} [u^{10}] \frac{d}{d x} [x^{4}]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 10$ .

$10 u^{9} \frac{d}{d x} [x^{4}]$

Step 1.3

Replace all occurrences of $u$ with $x^{4}$ .

$10 {(x^{4})}^{9} \frac{d}{d x} [x^{4}]$

Step 2

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

$10 {(x^{4})}^{9} (4 x^{3})$

Step 3

Combine terms.

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Step 3.1

Multiply the exponents in ${(x^{4})}^{9}$ .

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Step 3.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$10 x^{4 \cdot 9} (4 x^{3})$

Step 3.1.2

Multiply $4$ by $9$ .

$10 x^{36} (4 x^{3})$

Step 3.2

Multiply $4$ by $10$ .

$40 x^{36} x^{3}$

Step 3.3

Multiply $x^{36}$ by $x^{3}$ by adding the exponents.

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Step 3.3.1

Move $x^{3}$ .

$40 (x^{3} x^{36})$

Step 3.3.2

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

$40 x^{3 + 36}$

Step 3.3.3

Add $3$ and $36$ .

$40 x^{39}$