Calculus Examples

$\frac{\ln (7 x)}{x^{5}}$

Step 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 $f (x) = \ln (7 x)$ and $g (x) = x^{5}$ .

$\frac{x^{5} \frac{d}{d x} [\ln (7 x)] - \ln (7 x) \frac{d}{d x} [x^{5}]}{{(x^{5})}^{2}}$

Step 2

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

Tap for more steps...

Step 2.1

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

$\frac{x^{5} \frac{d}{d x} [\ln (7 x)] - \ln (7 x) \frac{d}{d x} [x^{5}]}{x^{5 \cdot 2}}$

Step 2.2

Multiply $5$ by $2$ .

$\frac{x^{5} \frac{d}{d x} [\ln (7 x)] - \ln (7 x) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = 7 x$ .

Tap for more steps...

Step 3.1

To apply the Chain Rule, set $u$ as $7 x$ .

$\frac{x^{5} (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [7 x]) - \ln (7 x) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 3.2

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

$\frac{x^{5} (\frac{1}{u} \frac{d}{d x} [7 x]) - \ln (7 x) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 3.3

Replace all occurrences of $u$ with $7 x$ .

$\frac{x^{5} (\frac{1}{7 x} \frac{d}{d x} [7 x]) - \ln (7 x) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 4

Differentiate.

Tap for more steps...

Step 4.1

Combine $\frac{1}{7 x}$ and $x^{5}$ .

$\frac{\frac{x^{5}}{7 x} \frac{d}{d x} [7 x] - \ln (7 x) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 4.2

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

Tap for more steps...

Step 4.2.1

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

$\frac{\frac{x \cdot x^{4}}{7 x} \frac{d}{d x} [7 x] - \ln (7 x) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 4.2.2

Cancel the common factors.

Tap for more steps...

Step 4.2.2.1

Factor $x$ out of $7 x$ .

$\frac{\frac{x \cdot x^{4}}{x \cdot 7} \frac{d}{d x} [7 x] - \ln (7 x) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 4.2.2.2

Cancel the common factor.

$\frac{\frac{x \cdot x^{4}}{x \cdot 7} \frac{d}{d x} [7 x] - \ln (7 x) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 4.2.2.3

Rewrite the expression.

$\frac{\frac{x^{4}}{7} \frac{d}{d x} [7 x] - \ln (7 x) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 4.3

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$\frac{\frac{x^{4}}{7} (7 \frac{d}{d x} [x]) - \ln (7 x) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 4.4

Simplify terms.

Tap for more steps...

Step 4.4.1

Combine $7$ and $\frac{x^{4}}{7}$ .

$\frac{\frac{7 x^{4}}{7} \frac{d}{d x} [x] - \ln (7 x) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 4.4.2

Cancel the common factor of $7$ .

Tap for more steps...

Step 4.4.2.1

Cancel the common factor.

$\frac{\frac{7 x^{4}}{7} \frac{d}{d x} [x] - \ln (7 x) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 4.4.2.2

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

$\frac{x^{4} \frac{d}{d x} [x] - \ln (7 x) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 4.5

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

$\frac{x^{4} \cdot 1 - \ln (7 x) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 4.6

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

$\frac{x^{4} - \ln (7 x) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 4.7

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

$\frac{x^{4} - \ln (7 x) (5 x^{4})}{x^{10}}$

Step 4.8

Simplify with factoring out.

Tap for more steps...

Step 4.8.1

Multiply $5$ by $- 1$ .

$\frac{x^{4} - 5 \ln (7 x) x^{4}}{x^{10}}$

Step 4.8.2

Factor $x^{4}$ out of $x^{4} - 5 \ln (7 x) x^{4}$ .

Tap for more steps...

Step 4.8.2.1

Multiply by $1$ .

$\frac{x^{4} \cdot 1 - 5 \ln (7 x) x^{4}}{x^{10}}$

Step 4.8.2.2

Factor $x^{4}$ out of $- 5 \ln (7 x) x^{4}$ .

$\frac{x^{4} \cdot 1 + x^{4} (- 5 \ln (7 x))}{x^{10}}$

Step 4.8.2.3

Factor $x^{4}$ out of $x^{4} \cdot 1 + x^{4} (- 5 \ln (7 x))$ .

$\frac{x^{4} (1 - 5 \ln (7 x))}{x^{10}}$

Step 5

Cancel the common factors.

Tap for more steps...

Step 5.1

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

$\frac{x^{4} (1 - 5 \ln (7 x))}{x^{4} x^{6}}$

Step 5.2

Cancel the common factor.

$\frac{x^{4} (1 - 5 \ln (7 x))}{x^{4} x^{6}}$

Step 5.3

Rewrite the expression.

$\frac{1 - 5 \ln (7 x)}{x^{6}}$

Step 6

Simplify each term.

Tap for more steps...

Step 6.1

Simplify $- 5 \ln (7 x)$ by moving $5$ inside the logarithm.

$\frac{1 - \ln ({(7 x)}^{5})}{x^{6}}$

Step 6.2

Apply the product rule to $7 x$ .

$\frac{1 - \ln (7^{5} x^{5})}{x^{6}}$

Step 6.3

Raise $7$ to the power of $5$ .

$\frac{1 - \ln (16807 x^{5})}{x^{6}}$