Calculus Examples

$15 x^{3}$

Step 1

Find the first derivative.

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Step 1.1

Since $15$ is constant with respect to $x$ , the derivative of $15 x^{3}$ with respect to $x$ is $15 \frac{d}{d x} [x^{3}]$ .

$15 \frac{d}{d x} [x^{3}]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$15 (3 x^{2})$

Step 1.3

Multiply $3$ by $15$ .

$f' (x) = 45 x^{2}$

Step 2

Find the second derivative.

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Step 2.1

Since $45$ is constant with respect to $x$ , the derivative of $45 x^{2}$ with respect to $x$ is $45 \frac{d}{d x} [x^{2}]$ .

$45 \frac{d}{d x} [x^{2}]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$45 (2 x)$

Step 2.3

Multiply $2$ by $45$ .

$f'' (x) = 90 x$

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