Calculus Examples

$y = {(3 x^{2})}^{4}$

Step 1

Simplify with factoring out.

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Step 1.1

Factor $3$ out of $3 x^{2}$ .

$\frac{d}{d x} [{(3 (x^{2}))}^{4}]$

Step 1.2

Simplify the expression.

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Step 1.2.1

Apply the product rule to $3 (x^{2})$ .

$\frac{d}{d x} [3^{4} {(x^{2})}^{4}]$

Step 1.2.2

Raise $3$ to the power of $4$ .

$\frac{d}{d x} [81 {(x^{2})}^{4}]$

Step 1.2.3

Multiply the exponents in ${(x^{2})}^{4}$ .

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Step 1.2.3.1

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

$\frac{d}{d x} [81 x^{2 \cdot 4}]$

Step 1.2.3.2

Multiply $2$ by $4$ .

$\frac{d}{d x} [81 x^{8}]$

Step 2

Since $81$ is constant with respect to $x$ , the derivative of $81 x^{8}$ with respect to $x$ is $81 \frac{d}{d x} [x^{8}]$ .

$81 \frac{d}{d x} [x^{8}]$

Step 3

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

$81 (8 x^{7})$

Step 4

Multiply $8$ by $81$ .

$648 x^{7}$