Calculus Examples

$f (x) = x^{2} g (x)$

Step 1

Find the first derivative.

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Step 1.1

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

$\frac{d}{d x} [g (x^{2} x^{1})]$

Step 1.2

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

$\frac{d}{d x} [g x^{2 + 1}]$

Step 1.3

Add $2$ and $1$ .

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

Step 1.4

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

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

Step 1.5

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

$g (3 x^{2})$

Step 1.6

Reorder.

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Step 1.6.1

Move $3$ to the left of $g$ .

$3 \cdot g x^{2}$

Step 1.6.2

Reorder the factors of $3 g x^{2}$ .

$f' (x) = 3 x^{2} g$

Step 2

Find the second derivative.

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Step 2.1

Since $3 g$ is constant with respect to $x$ , the derivative of $3 x^{2} g$ with respect to $x$ is $3 g \frac{d}{d x} [x^{2}]$ .

$3 g \frac{d}{d x} [x^{2}]$

Step 2.2

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

$3 g (2 x)$

Step 2.3

Multiply $2$ by $3$ .

$f'' (x) = 6 g x$

Step 3

The second derivative of $f (x)$ with respect to $x$ is $6 g x$ .

$6 g x$