Calculus Examples

$y = \frac{\ln (x)}{x^{13}}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{\ln (x)}{x^{13}})$

Step 2

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

$y'$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

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 $g (x) = x^{13}$ .

$\frac{x^{13} \frac{d}{d x} [\ln (x)] - \ln (x) \frac{d}{d x} [x^{13}]}{{(x^{13})}^{2}}$

Step 3.2

Multiply the exponents in ${(x^{13})}^{2}$ .

Tap for more steps...

Step 3.2.1

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

$\frac{x^{13} \frac{d}{d x} [\ln (x)] - \ln (x) \frac{d}{d x} [x^{13}]}{x^{13 \cdot 2}}$

Step 3.2.2

Multiply $13$ by $2$ .

$\frac{x^{13} \frac{d}{d x} [\ln (x)] - \ln (x) \frac{d}{d x} [x^{13}]}{x^{26}}$

Step 3.3

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

$\frac{x^{13} \frac{1}{x} - \ln (x) \frac{d}{d x} [x^{13}]}{x^{26}}$

Step 3.4

Differentiate using the Power Rule.

Tap for more steps...

Step 3.4.1

Combine $x^{13}$ and $\frac{1}{x}$ .

$\frac{\frac{x^{13}}{x} - \ln (x) \frac{d}{d x} [x^{13}]}{x^{26}}$

Step 3.4.2

Cancel the common factor of $x^{13}$ and $x$ .

Tap for more steps...

Step 3.4.2.1

Factor $x$ out of $x^{13}$ .

$\frac{\frac{x \cdot x^{12}}{x} - \ln (x) \frac{d}{d x} [x^{13}]}{x^{26}}$

Step 3.4.2.2

Cancel the common factors.

Tap for more steps...

Step 3.4.2.2.1

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

$\frac{\frac{x \cdot x^{12}}{x^{1}} - \ln (x) \frac{d}{d x} [x^{13}]}{x^{26}}$

Step 3.4.2.2.2

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

$\frac{\frac{x \cdot x^{12}}{x \cdot 1} - \ln (x) \frac{d}{d x} [x^{13}]}{x^{26}}$

Step 3.4.2.2.3

Cancel the common factor.

$\frac{\frac{x \cdot x^{12}}{x \cdot 1} - \ln (x) \frac{d}{d x} [x^{13}]}{x^{26}}$

Step 3.4.2.2.4

Rewrite the expression.

$\frac{\frac{x^{12}}{1} - \ln (x) \frac{d}{d x} [x^{13}]}{x^{26}}$

Step 3.4.2.2.5

Divide $x^{12}$ by $1$ .

$\frac{x^{12} - \ln (x) \frac{d}{d x} [x^{13}]}{x^{26}}$

Step 3.4.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 13$ .

$\frac{x^{12} - \ln (x) (13 x^{12})}{x^{26}}$

Step 3.4.4

Simplify with factoring out.

Tap for more steps...

Step 3.4.4.1

Multiply $13$ by $- 1$ .

$\frac{x^{12} - 13 \ln (x) x^{12}}{x^{26}}$

Step 3.4.4.2

Factor $x^{12}$ out of $x^{12} - 13 \ln (x) x^{12}$ .

Tap for more steps...

Step 3.4.4.2.1

Multiply by $1$ .

$\frac{x^{12} \cdot 1 - 13 \ln (x) x^{12}}{x^{26}}$

Step 3.4.4.2.2

Factor $x^{12}$ out of $- 13 \ln (x) x^{12}$ .

$\frac{x^{12} \cdot 1 + x^{12} (- 13 \ln (x))}{x^{26}}$

Step 3.4.4.2.3

Factor $x^{12}$ out of $x^{12} \cdot 1 + x^{12} (- 13 \ln (x))$ .

$\frac{x^{12} (1 - 13 \ln (x))}{x^{26}}$

Step 3.5

Cancel the common factors.

Tap for more steps...

Step 3.5.1

Factor $x^{12}$ out of $x^{26}$ .

$\frac{x^{12} (1 - 13 \ln (x))}{x^{12} x^{14}}$

Step 3.5.2

Cancel the common factor.

$\frac{x^{12} (1 - 13 \ln (x))}{x^{12} x^{14}}$

Step 3.5.3

Rewrite the expression.

$\frac{1 - 13 \ln (x)}{x^{14}}$

Step 3.6

Simplify $- 13 \ln (x)$ by moving $13$ inside the logarithm.

$\frac{1 - \ln (x^{13})}{x^{14}}$

Step 4

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

$y' = \frac{1 - \ln (x^{13})}{x^{14}}$

Step 5

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

$\frac{d y}{d x} = \frac{1 - \ln (x^{13})}{x^{14}}$