Calculus Examples

$y = \frac{x^{7}}{7} \cdot \ln (x) - \frac{x^{7}}{49}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{x^{7}}{7} \cdot \ln (x) - \frac{x^{7}}{49})$

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

By the Sum Rule, the derivative of $\frac{x^{7}}{7} \cdot \ln (x) - \frac{x^{7}}{49}$ with respect to $x$ is $\frac{d}{d x} [\frac{x^{7}}{7} \cdot \ln (x)] + \frac{d}{d x} [- \frac{x^{7}}{49}]$ .

$\frac{d}{d x} [\frac{x^{7}}{7} \cdot \ln (x)] + \frac{d}{d x} [- \frac{x^{7}}{49}]$

Step 3.2

Evaluate $\frac{d}{d x} [\frac{x^{7}}{7} \cdot \ln (x)]$ .

Tap for more steps...

Step 3.2.1

Combine $\frac{x^{7}}{7}$ and $\ln (x)$ .

$\frac{d}{d x} [\frac{x^{7} \ln (x)}{7}] + \frac{d}{d x} [- \frac{x^{7}}{49}]$

Step 3.2.2

Since $\frac{1}{7}$ is constant with respect to $x$ , the derivative of $\frac{x^{7} \ln (x)}{7}$ with respect to $x$ is $\frac{1}{7} \frac{d}{d x} [x^{7} \ln (x)]$ .

$\frac{1}{7} \frac{d}{d x} [x^{7} \ln (x)] + \frac{d}{d x} [- \frac{x^{7}}{49}]$

Step 3.2.3

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

$\frac{1}{7} (x^{7} \frac{d}{d x} [\ln (x)] + \ln (x) \frac{d}{d x} [x^{7}]) + \frac{d}{d x} [- \frac{x^{7}}{49}]$

Step 3.2.4

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

$\frac{1}{7} (x^{7} \frac{1}{x} + \ln (x) \frac{d}{d x} [x^{7}]) + \frac{d}{d x} [- \frac{x^{7}}{49}]$

Step 3.2.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 7$ .

$\frac{1}{7} (x^{7} \frac{1}{x} + \ln (x) (7 x^{6})) + \frac{d}{d x} [- \frac{x^{7}}{49}]$

Step 3.2.6

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

$\frac{1}{7} (\frac{x^{7}}{x} + \ln (x) (7 x^{6})) + \frac{d}{d x} [- \frac{x^{7}}{49}]$

Step 3.2.7

Cancel the common factor of $x^{7}$ and $x$ .

Tap for more steps...

Step 3.2.7.1

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

$\frac{1}{7} (\frac{x \cdot x^{6}}{x} + \ln (x) (7 x^{6})) + \frac{d}{d x} [- \frac{x^{7}}{49}]$

Step 3.2.7.2

Cancel the common factors.

Tap for more steps...

Step 3.2.7.2.1

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

$\frac{1}{7} (\frac{x \cdot x^{6}}{x^{1}} + \ln (x) (7 x^{6})) + \frac{d}{d x} [- \frac{x^{7}}{49}]$

Step 3.2.7.2.2

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

$\frac{1}{7} (\frac{x \cdot x^{6}}{x \cdot 1} + \ln (x) (7 x^{6})) + \frac{d}{d x} [- \frac{x^{7}}{49}]$

Step 3.2.7.2.3

Cancel the common factor.

$\frac{1}{7} (\frac{x \cdot x^{6}}{x \cdot 1} + \ln (x) (7 x^{6})) + \frac{d}{d x} [- \frac{x^{7}}{49}]$

Step 3.2.7.2.4

Rewrite the expression.

$\frac{1}{7} (\frac{x^{6}}{1} + \ln (x) (7 x^{6})) + \frac{d}{d x} [- \frac{x^{7}}{49}]$

Step 3.2.7.2.5

Divide $x^{6}$ by $1$ .

$\frac{1}{7} (x^{6} + \ln (x) (7 x^{6})) + \frac{d}{d x} [- \frac{x^{7}}{49}]$

Step 3.3

Evaluate $\frac{d}{d x} [- \frac{x^{7}}{49}]$ .

Tap for more steps...

Step 3.3.1

Since $- \frac{1}{49}$ is constant with respect to $x$ , the derivative of $- \frac{x^{7}}{49}$ with respect to $x$ is $- \frac{1}{49} \frac{d}{d x} [x^{7}]$ .

$\frac{1}{7} (x^{6} + \ln (x) (7 x^{6})) - \frac{1}{49} \frac{d}{d x} [x^{7}]$

Step 3.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 7$ .

$\frac{1}{7} (x^{6} + \ln (x) (7 x^{6})) - \frac{1}{49} (7 x^{6})$

Step 3.3.3

Multiply $7$ by $- 1$ .

$\frac{1}{7} (x^{6} + \ln (x) (7 x^{6})) - 7 (\frac{1}{49}) x^{6}$

Step 3.3.4

Combine $- 7$ and $\frac{1}{49}$ .

$\frac{1}{7} (x^{6} + \ln (x) (7 x^{6})) + \frac{- 7}{49} x^{6}$

Step 3.3.5

Combine $\frac{- 7}{49}$ and $x^{6}$ .

$\frac{1}{7} (x^{6} + \ln (x) (7 x^{6})) + \frac{- 7 x^{6}}{49}$

Step 3.3.6

Cancel the common factor of $- 7$ and $49$ .

Tap for more steps...

Step 3.3.6.1

Factor $7$ out of $- 7 x^{6}$ .

$\frac{1}{7} (x^{6} + \ln (x) (7 x^{6})) + \frac{7 (- x^{6})}{49}$

Step 3.3.6.2

Cancel the common factors.

Tap for more steps...

Step 3.3.6.2.1

Factor $7$ out of $49$ .

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

Step 3.3.6.2.2

Cancel the common factor.

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

Step 3.3.6.2.3

Rewrite the expression.

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

Step 3.3.7

Move the negative in front of the fraction.

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

Step 3.4

Simplify.

Tap for more steps...

Step 3.4.1

Apply the distributive property.

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

Step 3.4.2

Combine terms.

Tap for more steps...

Step 3.4.2.1

Combine $\ln (x)$ and $\frac{1}{7}$ .

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

Step 3.4.2.2

Combine $7$ and $\frac{\ln (x)}{7}$ .

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

Step 3.4.2.3

Combine $\frac{7 \ln (x)}{7}$ and $x^{6}$ .

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

Step 3.4.2.4

Cancel the common factor of $7$ .

Tap for more steps...

Step 3.4.2.4.1

Cancel the common factor.

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

Step 3.4.2.4.2

Divide $\ln (x) x^{6}$ by $1$ .

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

Step 3.4.2.5

To write $\frac{1}{7} x^{6}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

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

Step 3.4.2.6

Combine $\frac{1}{7} x^{6}$ and $\frac{7}{7}$ .

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

Step 3.4.2.7

Combine the numerators over the common denominator.

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

Step 3.4.2.8

Combine $\frac{1}{7}$ and $x^{6}$ .

$\ln (x) x^{6} + \frac{\frac{x^{6}}{7} \cdot 7 - x^{6}}{7}$

Step 3.4.2.9

Combine $\frac{x^{6}}{7}$ and $7$ .

$\ln (x) x^{6} + \frac{\frac{x^{6} \cdot 7}{7} - x^{6}}{7}$

Step 3.4.2.10

Move $7$ to the left of $x^{6}$ .

$\ln (x) x^{6} + \frac{\frac{7 \cdot x^{6}}{7} - x^{6}}{7}$

Step 3.4.2.11

Cancel the common factor of $7$ .

Tap for more steps...

Step 3.4.2.11.1

Cancel the common factor.

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

Step 3.4.2.11.2

Divide $x^{6}$ by $1$ .

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

Step 3.4.2.12

Subtract $x^{6}$ from $x^{6}$ .

$\ln (x) x^{6} + \frac{0}{7}$

Step 3.4.2.13

Cancel the common factor of $0$ and $7$ .

Tap for more steps...

Step 3.4.2.13.1

Factor $7$ out of $0$ .

$\ln (x) x^{6} + \frac{7 (0)}{7}$

Step 3.4.2.13.2

Cancel the common factors.

Tap for more steps...

Step 3.4.2.13.2.1

Factor $7$ out of $7$ .

$\ln (x) x^{6} + \frac{7 \cdot 0}{7 \cdot 1}$

Step 3.4.2.13.2.2

Cancel the common factor.

$\ln (x) x^{6} + \frac{7 \cdot 0}{7 \cdot 1}$

Step 3.4.2.13.2.3

Rewrite the expression.

$\ln (x) x^{6} + \frac{0}{1}$

Step 3.4.2.13.2.4

Divide $0$ by $1$ .

$\ln (x) x^{6} + 0$

Step 3.4.2.14

Add $\ln (x) x^{6}$ and $0$ .

$\ln (x) x^{6}$

Step 3.4.3

Reorder the factors of $\ln (x) x^{6}$ .

$x^{6} \ln (x)$

Step 4

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

$y' = x^{6} \ln (x)$

Step 5

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

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