Calculus Examples

$y = 3 \ln (x) - \frac{5}{x^{4}}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (3 \ln (x) - \frac{5}{x^{4}})$

Step 2

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

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $3 \ln (x) - \frac{5}{x^{4}}$ with respect to $x$ is $\frac{d}{d x} [3 \ln (x)] + \frac{d}{d x} [- \frac{5}{x^{4}}]$ .

$\frac{d}{d x} [3 \ln (x)] + \frac{d}{d x} [- \frac{5}{x^{4}}]$

Step 3.2

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

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Step 3.2.1

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

$3 \frac{d}{d x} [\ln (x)] + \frac{d}{d x} [- \frac{5}{x^{4}}]$

Step 3.2.2

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

$3 \frac{1}{x} + \frac{d}{d x} [- \frac{5}{x^{4}}]$

Step 3.2.3

Combine $3$ and $\frac{1}{x}$ .

$\frac{3}{x} + \frac{d}{d x} [- \frac{5}{x^{4}}]$

Step 3.3

Evaluate $\frac{d}{d x} [- \frac{5}{x^{4}}]$ .

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Step 3.3.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- \frac{5}{x^{4}}$ with respect to $x$ is $- 5 \frac{d}{d x} [\frac{1}{x^{4}}]$ .

$\frac{3}{x} - 5 \frac{d}{d x} [\frac{1}{x^{4}}]$

Step 3.3.2

Rewrite $\frac{1}{x^{4}}$ as ${(x^{4})}^{- 1}$ .

$\frac{3}{x} - 5 \frac{d}{d x} [{(x^{4})}^{- 1}]$

Step 3.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{4}$ .

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Step 3.3.3.1

To apply the Chain Rule, set $u$ as $x^{4}$ .

$\frac{3}{x} - 5 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{4}])$

Step 3.3.3.2

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

$\frac{3}{x} - 5 (- u^{- 2} \frac{d}{d x} [x^{4}])$

Step 3.3.3.3

Replace all occurrences of $u$ with $x^{4}$ .

$\frac{3}{x} - 5 (- {(x^{4})}^{- 2} \frac{d}{d x} [x^{4}])$

Step 3.3.4

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

$\frac{3}{x} - 5 (- {(x^{4})}^{- 2} (4 x^{3}))$

Step 3.3.5

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

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Step 3.3.5.1

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

$\frac{3}{x} - 5 (- x^{4 \cdot - 2} (4 x^{3}))$

Step 3.3.5.2

Multiply $4$ by $- 2$ .

$\frac{3}{x} - 5 (- x^{- 8} (4 x^{3}))$

Step 3.3.6

Multiply $4$ by $- 1$ .

$\frac{3}{x} - 5 (- 4 x^{- 8} x^{3})$

Step 3.3.7

Multiply $x^{- 8}$ by $x^{3}$ by adding the exponents.

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Step 3.3.7.1

Move $x^{3}$ .

$\frac{3}{x} - 5 (- 4 (x^{3} x^{- 8}))$

Step 3.3.7.2

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

$\frac{3}{x} - 5 (- 4 x^{3 - 8})$

Step 3.3.7.3

Subtract $8$ from $3$ .

$\frac{3}{x} - 5 (- 4 x^{- 5})$

Step 3.3.8

Multiply $- 4$ by $- 5$ .

$\frac{3}{x} + 20 x^{- 5}$

Step 3.4

Simplify.

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Step 3.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{3}{x} + 20 \frac{1}{x^{5}}$

Step 3.4.2

Combine $20$ and $\frac{1}{x^{5}}$ .

$\frac{3}{x} + \frac{20}{x^{5}}$

Step 4

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

$y' = \frac{3}{x} + \frac{20}{x^{5}}$

Step 5

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

$\frac{d y}{d x} = \frac{3}{x} + \frac{20}{x^{5}}$