Calculus Examples

$y = \frac{x^{5}}{10} \cdot \ln (x) - \frac{x^{5}}{50}$

Step 1

This derivative could not be completed using the quotient rule. Mathway will use another method.

Step 2

By the Sum Rule, the derivative of $\frac{x^{5}}{10} \cdot \ln (x) - \frac{x^{5}}{50}$ with respect to $x$ is $\frac{d}{d x} [\frac{x^{5}}{10} \cdot \ln (x)] + \frac{d}{d x} [- \frac{x^{5}}{50}]$ .

$\frac{d}{d x} [\frac{x^{5}}{10} \cdot \ln (x)] + \frac{d}{d x} [- \frac{x^{5}}{50}]$

Step 3

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

Tap for more steps...

Step 3.1

Combine $\frac{x^{5}}{10}$ and $\ln (x)$ .

$\frac{d}{d x} [\frac{x^{5} \ln (x)}{10}] + \frac{d}{d x} [- \frac{x^{5}}{50}]$

Step 3.2

Since $\frac{1}{10}$ is constant with respect to $x$ , the derivative of $\frac{x^{5} \ln (x)}{10}$ with respect to $x$ is $\frac{1}{10} \frac{d}{d x} [x^{5} \ln (x)]$ .

$\frac{1}{10} \frac{d}{d x} [x^{5} \ln (x)] + \frac{d}{d x} [- \frac{x^{5}}{50}]$

Step 3.3

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

$\frac{1}{10} (x^{5} \frac{d}{d x} [\ln (x)] + \ln (x) \frac{d}{d x} [x^{5}]) + \frac{d}{d x} [- \frac{x^{5}}{50}]$

Step 3.4

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

$\frac{1}{10} (x^{5} \frac{1}{x} + \ln (x) \frac{d}{d x} [x^{5}]) + \frac{d}{d x} [- \frac{x^{5}}{50}]$

Step 3.5

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

$\frac{1}{10} (x^{5} \frac{1}{x} + \ln (x) (5 x^{4})) + \frac{d}{d x} [- \frac{x^{5}}{50}]$

Step 3.6

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

$\frac{1}{10} (\frac{x^{5}}{x} + \ln (x) (5 x^{4})) + \frac{d}{d x} [- \frac{x^{5}}{50}]$

Step 3.7

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

Tap for more steps...

Step 3.7.1

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

$\frac{1}{10} (\frac{x \cdot x^{4}}{x} + \ln (x) (5 x^{4})) + \frac{d}{d x} [- \frac{x^{5}}{50}]$

Step 3.7.2

Cancel the common factors.

Tap for more steps...

Step 3.7.2.1

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

$\frac{1}{10} (\frac{x \cdot x^{4}}{x^{1}} + \ln (x) (5 x^{4})) + \frac{d}{d x} [- \frac{x^{5}}{50}]$

Step 3.7.2.2

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

$\frac{1}{10} (\frac{x \cdot x^{4}}{x \cdot 1} + \ln (x) (5 x^{4})) + \frac{d}{d x} [- \frac{x^{5}}{50}]$

Step 3.7.2.3

Cancel the common factor.

$\frac{1}{10} (\frac{x \cdot x^{4}}{x \cdot 1} + \ln (x) (5 x^{4})) + \frac{d}{d x} [- \frac{x^{5}}{50}]$

Step 3.7.2.4

Rewrite the expression.

$\frac{1}{10} (\frac{x^{4}}{1} + \ln (x) (5 x^{4})) + \frac{d}{d x} [- \frac{x^{5}}{50}]$

Step 3.7.2.5

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

$\frac{1}{10} (x^{4} + \ln (x) (5 x^{4})) + \frac{d}{d x} [- \frac{x^{5}}{50}]$

Step 4

Evaluate $\frac{d}{d x} [- \frac{x^{5}}{50}]$ .

Tap for more steps...

Step 4.1

Since $- \frac{1}{50}$ is constant with respect to $x$ , the derivative of $- \frac{x^{5}}{50}$ with respect to $x$ is $- \frac{1}{50} \frac{d}{d x} [x^{5}]$ .

$\frac{1}{10} (x^{4} + \ln (x) (5 x^{4})) - \frac{1}{50} \frac{d}{d x} [x^{5}]$

Step 4.2

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

$\frac{1}{10} (x^{4} + \ln (x) (5 x^{4})) - \frac{1}{50} (5 x^{4})$

Step 4.3

Multiply $5$ by $- 1$ .

$\frac{1}{10} (x^{4} + \ln (x) (5 x^{4})) - 5 (\frac{1}{50}) x^{4}$

Step 4.4

Combine $- 5$ and $\frac{1}{50}$ .

$\frac{1}{10} (x^{4} + \ln (x) (5 x^{4})) + \frac{- 5}{50} x^{4}$

Step 4.5

Combine $\frac{- 5}{50}$ and $x^{4}$ .

$\frac{1}{10} (x^{4} + \ln (x) (5 x^{4})) + \frac{- 5 x^{4}}{50}$

Step 4.6

Cancel the common factor of $- 5$ and $50$ .

Tap for more steps...

Step 4.6.1

Factor $5$ out of $- 5 x^{4}$ .

$\frac{1}{10} (x^{4} + \ln (x) (5 x^{4})) + \frac{5 (- x^{4})}{50}$

Step 4.6.2

Cancel the common factors.

Tap for more steps...

Step 4.6.2.1

Factor $5$ out of $50$ .

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

Step 4.6.2.2

Cancel the common factor.

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

Step 4.6.2.3

Rewrite the expression.

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

Step 4.7

Move the negative in front of the fraction.

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

Step 5

Simplify.

Tap for more steps...

Step 5.1

Apply the distributive property.

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

Step 5.2

Combine terms.

Tap for more steps...

Step 5.2.1

Combine $\ln (x)$ and $\frac{1}{10}$ .

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

Step 5.2.2

Combine $5$ and $\frac{\ln (x)}{10}$ .

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

Step 5.2.3

Combine $\frac{5 \ln (x)}{10}$ and $x^{4}$ .

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

Step 5.2.4

Cancel the common factor of $5$ and $10$ .

Tap for more steps...

Step 5.2.4.1

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

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

Step 5.2.4.2

Cancel the common factors.

Tap for more steps...

Step 5.2.4.2.1

Factor $5$ out of $10$ .

$\frac{1}{10} x^{4} + \frac{5 (\ln (x) x^{4})}{5 \cdot 2} - \frac{x^{4}}{10}$

Step 5.2.4.2.2

Cancel the common factor.

$\frac{1}{10} x^{4} + \frac{5 (\ln (x) x^{4})}{5 \cdot 2} - \frac{x^{4}}{10}$

Step 5.2.4.2.3

Rewrite the expression.

$\frac{1}{10} x^{4} + \frac{\ln (x) x^{4}}{2} - \frac{x^{4}}{10}$

Step 5.2.5

To write $\frac{1}{10} x^{4}$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$\frac{\ln (x) x^{4}}{2} + \frac{1}{10} x^{4} \cdot \frac{10}{10} - \frac{x^{4}}{10}$

Step 5.2.6

Combine $\frac{1}{10} x^{4}$ and $\frac{10}{10}$ .

$\frac{\ln (x) x^{4}}{2} + \frac{\frac{1}{10} x^{4} \cdot 10}{10} - \frac{x^{4}}{10}$

Step 5.2.7

Combine the numerators over the common denominator.

$\frac{\ln (x) x^{4}}{2} + \frac{\frac{1}{10} x^{4} \cdot 10 - x^{4}}{10}$

Step 5.2.8

Combine $\frac{1}{10}$ and $x^{4}$ .

$\frac{\ln (x) x^{4}}{2} + \frac{\frac{x^{4}}{10} \cdot 10 - x^{4}}{10}$

Step 5.2.9

Combine $\frac{x^{4}}{10}$ and $10$ .

$\frac{\ln (x) x^{4}}{2} + \frac{\frac{x^{4} \cdot 10}{10} - x^{4}}{10}$

Step 5.2.10

Move $10$ to the left of $x^{4}$ .

$\frac{\ln (x) x^{4}}{2} + \frac{\frac{10 \cdot x^{4}}{10} - x^{4}}{10}$

Step 5.2.11

Cancel the common factor of $10$ .

Tap for more steps...

Step 5.2.11.1

Cancel the common factor.

$\frac{\ln (x) x^{4}}{2} + \frac{\frac{10 x^{4}}{10} - x^{4}}{10}$

Step 5.2.11.2

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

$\frac{\ln (x) x^{4}}{2} + \frac{x^{4} - x^{4}}{10}$

Step 5.2.12

Subtract $x^{4}$ from $x^{4}$ .

$\frac{\ln (x) x^{4}}{2} + \frac{0}{10}$

Step 5.2.13

Cancel the common factor of $0$ and $10$ .

Tap for more steps...

Step 5.2.13.1

Factor $10$ out of $0$ .

$\frac{\ln (x) x^{4}}{2} + \frac{10 (0)}{10}$

Step 5.2.13.2

Cancel the common factors.

Tap for more steps...

Step 5.2.13.2.1

Factor $10$ out of $10$ .

$\frac{\ln (x) x^{4}}{2} + \frac{10 \cdot 0}{10 \cdot 1}$

Step 5.2.13.2.2

Cancel the common factor.

$\frac{\ln (x) x^{4}}{2} + \frac{10 \cdot 0}{10 \cdot 1}$

Step 5.2.13.2.3

Rewrite the expression.

$\frac{\ln (x) x^{4}}{2} + \frac{0}{1}$

Step 5.2.13.2.4

Divide $0$ by $1$ .

$\frac{\ln (x) x^{4}}{2} + 0$

Step 5.2.14

Add $\frac{\ln (x) x^{4}}{2}$ and $0$ .

$\frac{\ln (x) x^{4}}{2}$

Step 5.3

Reorder terms.

$\frac{x^{4} \ln (x)}{2}$