Calculus Examples

$y = {(x + 4 \ln (x))}^{4}$

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} ({(x + 4 \ln (x))}^{4})$

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Differentiate the right side of the equation.

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

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To apply the Chain Rule, set $u$ as $x + 4 \ln (x)$ .

$\frac{d}{d u} [u^{4}] \frac{d}{d x} [x + 4 \ln (x)]$

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

$4 u^{3} \frac{d}{d x} [x + 4 \ln (x)]$

Replace all occurrences of $u$ with $x + 4 \ln (x)$ .

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

Differentiate.

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By the Sum Rule, the derivative of $x + 4 \ln (x)$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [4 \ln (x)]$ .

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

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

$4 {(x + 4 \ln (x))}^{3} (1 + \frac{d}{d x} [4 \ln (x)])$

Since $4$ is constant with respect to $x$ , the derivative of $4 \ln (x)$ with respect to $x$ is $4 \frac{d}{d x} [\ln (x)]$ .

$4 {(x + 4 \ln (x))}^{3} (1 + 4 \frac{d}{d x} [\ln (x)])$

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$4 {(x + 4 \ln (x))}^{3} (1 + 4 \frac{1}{x})$

Combine $4$ and $\frac{1}{x}$ .

$4 {(x + 4 \ln (x))}^{3} (1 + \frac{4}{x})$

Reform the equation by setting the left side equal to the right side.

$y' = 4 {(x + 4 \ln (x))}^{3} (1 + \frac{4}{x})$

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = 4 {(x + 4 \ln (x))}^{3} (1 + \frac{4}{x})$