Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

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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Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = \ln (x)$ and $g (x) = x^{5}$ .

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

Multiply the exponents in ${(x^{5})}^{2}$ .

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

$\frac{x^{5} \frac{d}{d x} [\ln (x)] - \ln (x) \frac{d}{d x} [x^{5}]}{x^{5 \cdot 2}}$

Multiply $5$ by $2$ .

$\frac{x^{5} \frac{d}{d x} [\ln (x)] - \ln (x) \frac{d}{d x} [x^{5}]}{x^{10}}$

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

$\frac{x^{5} \frac{1}{x} - \ln (x) \frac{d}{d x} [x^{5}]}{x^{10}}$

Differentiate using the Power Rule.

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

$\frac{\frac{x^{5}}{x} - \ln (x) \frac{d}{d x} [x^{5}]}{x^{10}}$

Cancel the common factor of $x^{5}$ and $x$ .

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Factor $x$ out of $x^{5}$ .

$\frac{\frac{x \cdot x^{4}}{x} - \ln (x) \frac{d}{d x} [x^{5}]}{x^{10}}$

Cancel the common factors.

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

$\frac{\frac{x \cdot x^{4}}{x^{1}} - \ln (x) \frac{d}{d x} [x^{5}]}{x^{10}}$

Factor $x$ out of $x^{1}$ .

$\frac{\frac{x \cdot x^{4}}{x \cdot 1} - \ln (x) \frac{d}{d x} [x^{5}]}{x^{10}}$

Cancel the common factor.

$\frac{\frac{x \cdot x^{4}}{x \cdot 1} - \ln (x) \frac{d}{d x} [x^{5}]}{x^{10}}$

Rewrite the expression.

$\frac{\frac{x^{4}}{1} - \ln (x) \frac{d}{d x} [x^{5}]}{x^{10}}$

Divide $x^{4}$ by $1$ .

$\frac{x^{4} - \ln (x) \frac{d}{d x} [x^{5}]}{x^{10}}$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$\frac{x^{4} - \ln (x) (5 x^{4})}{x^{10}}$

Simplify with factoring out.

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Multiply $5$ by $- 1$ .

$\frac{x^{4} - 5 \ln (x) x^{4}}{x^{10}}$

Factor $x^{4}$ out of $x^{4} - 5 \ln (x) x^{4}$ .

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Multiply by $1$ .

$\frac{x^{4} \cdot 1 - 5 \ln (x) x^{4}}{x^{10}}$

Factor $x^{4}$ out of $- 5 \ln (x) x^{4}$ .

$\frac{x^{4} \cdot 1 + x^{4} (- 5 \ln (x))}{x^{10}}$

Factor $x^{4}$ out of $x^{4} \cdot 1 + x^{4} (- 5 \ln (x))$ .

$\frac{x^{4} (1 - 5 \ln (x))}{x^{10}}$

Cancel the common factors.

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Factor $x^{4}$ out of $x^{10}$ .

$\frac{x^{4} (1 - 5 \ln (x))}{x^{4} x^{6}}$

Cancel the common factor.

$\frac{x^{4} (1 - 5 \ln (x))}{x^{4} x^{6}}$

Rewrite the expression.

$\frac{1 - 5 \ln (x)}{x^{6}}$

Simplify $- 5 \ln (x)$ by moving $5$ inside the logarithm.

$\frac{1 - \ln (x^{5})}{x^{6}}$

Step 4

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

$y' = \frac{1 - \ln (x^{5})}{x^{6}}$

Step 5

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

$\frac{d y}{d x} = \frac{1 - \ln (x^{5})}{x^{6}}$