Algebra Examples

$r^{3} \ln (x) - 2 \sqrt[3]{x}$

Step 1

By the Sum Rule, the derivative of $r^{3} \ln (x) - 2 \sqrt[3]{x}$ with respect to $r$ is $\frac{d}{d r} [r^{3} \ln (x)] + \frac{d}{d r} [- 2 \sqrt[3]{x}]$ .

$\frac{d}{d r} [r^{3} \ln (x)] + \frac{d}{d r} [- 2 \sqrt[3]{x}]$

Step 2

Evaluate $\frac{d}{d r} [r^{3} \ln (x)]$ .

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Step 2.1

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

$\ln (x) \frac{d}{d r} [r^{3}] + \frac{d}{d r} [- 2 \sqrt[3]{x}]$

Step 2.2

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

$\ln (x) (3 r^{2}) + \frac{d}{d r} [- 2 \sqrt[3]{x}]$

Step 2.3

Move $3$ to the left of $\ln (x)$ .

$3 \ln (x) r^{2} + \frac{d}{d r} [- 2 \sqrt[3]{x}]$

Step 3

Since $- 2 \sqrt[3]{x}$ is constant with respect to $r$ , the derivative of $- 2 \sqrt[3]{x}$ with respect to $r$ is $0$ .

$3 \ln (x) r^{2} + 0$

Step 4

Simplify.

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Step 4.1

Add $3 \ln (x) r^{2}$ and $0$ .

$3 \ln (x) r^{2}$

Step 4.2

Reorder the factors of $3 \ln (x) r^{2}$ .

$3 r^{2} \ln (x)$