Find dy/dx y=( natural log of x)/(x^8)

$y = \frac{\ln (x)}{x^{8}}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{\ln (x)}{x^{8}})$

Step 2

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

$y'$

Step 3

Differentiate the right side of the equation.

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 $g (x) = x^{8}$ .

$\frac{x^{8} \frac{d}{d x} [\ln (x)] - \ln (x) \frac{d}{d x} [x^{8}]}{{(x^{8})}^{2}}$

Multiply the exponents in ${(x^{8})}^{2}$ .

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

$\frac{x^{8} \frac{d}{d x} [\ln (x)] - \ln (x) \frac{d}{d x} [x^{8}]}{x^{8 \cdot 2}}$

Multiply $8$ by $2$ .

$\frac{x^{8} \frac{d}{d x} [\ln (x)] - \ln (x) \frac{d}{d x} [x^{8}]}{x^{16}}$

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

$\frac{x^{8} \frac{1}{x} - \ln (x) \frac{d}{d x} [x^{8}]}{x^{16}}$

Differentiate using the Power Rule.

Combine $x^{8}$ and $\frac{1}{x}$ .

$\frac{\frac{x^{8}}{x} - \ln (x) \frac{d}{d x} [x^{8}]}{x^{16}}$

Cancel the common factor of $x^{8}$ and $x$ .

Factor $x$ out of $x^{8}$ .

$\frac{\frac{x \cdot x^{7}}{x} - \ln (x) \frac{d}{d x} [x^{8}]}{x^{16}}$

Cancel the common factors.

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

$\frac{\frac{x \cdot x^{7}}{x^{1}} - \ln (x) \frac{d}{d x} [x^{8}]}{x^{16}}$

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

$\frac{\frac{x \cdot x^{7}}{x \cdot 1} - \ln (x) \frac{d}{d x} [x^{8}]}{x^{16}}$

Cancel the common factor.

$\frac{\frac{x \cdot x^{7}}{x \cdot 1} - \ln (x) \frac{d}{d x} [x^{8}]}{x^{16}}$

Rewrite the expression.

$\frac{\frac{x^{7}}{1} - \ln (x) \frac{d}{d x} [x^{8}]}{x^{16}}$

Divide $x^{7}$ by $1$ .

$\frac{x^{7} - \ln (x) \frac{d}{d x} [x^{8}]}{x^{16}}$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 8$ .

$\frac{x^{7} - \ln (x) (8 x^{7})}{x^{16}}$

Simplify with factoring out.

Multiply $8$ by $- 1$ .

$\frac{x^{7} - 8 \ln (x) x^{7}}{x^{16}}$

Factor $x^{7}$ out of $x^{7} - 8 \ln (x) x^{7}$ .

Multiply by $1$ .

$\frac{x^{7} \cdot 1 - 8 \ln (x) x^{7}}{x^{16}}$

Factor $x^{7}$ out of $- 8 \ln (x) x^{7}$ .

$\frac{x^{7} \cdot 1 + x^{7} (- 8 \ln (x))}{x^{16}}$

Factor $x^{7}$ out of $x^{7} \cdot 1 + x^{7} (- 8 \ln (x))$ .

$\frac{x^{7} (1 - 8 \ln (x))}{x^{16}}$

Cancel the common factors.

Factor $x^{7}$ out of $x^{16}$ .

$\frac{x^{7} (1 - 8 \ln (x))}{x^{7} x^{9}}$

Cancel the common factor.

$\frac{x^{7} (1 - 8 \ln (x))}{x^{7} x^{9}}$

Rewrite the expression.

$\frac{1 - 8 \ln (x)}{x^{9}}$

Simplify $- 8 \ln (x)$ by moving $8$ inside the logarithm.

$\frac{1 - \ln (x^{8})}{x^{9}}$

Step 4

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

$y' = \frac{1 - \ln (x^{8})}{x^{9}}$

Step 5

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

$\frac{d y}{d x} = \frac{1 - \ln (x^{8})}{x^{9}}$

Find dy/dx y=( natural log of x)/(x^8)

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