Calculus Examples

$y = \frac{8 x^{3}}{3} - 7 x$

Step 1

By the Sum Rule, the derivative of $\frac{8 x^{3}}{3} - 7 x$ with respect to $x$ is $\frac{d}{d x} [\frac{8 x^{3}}{3}] + \frac{d}{d x} [- 7 x]$ .

$\frac{d}{d x} [\frac{8 x^{3}}{3}] + \frac{d}{d x} [- 7 x]$

Step 2

Evaluate $\frac{d}{d x} [\frac{8 x^{3}}{3}]$ .

Tap for more steps...

Step 2.1

Since $\frac{8}{3}$ is constant with respect to $x$ , the derivative of $\frac{8 x^{3}}{3}$ with respect to $x$ is $\frac{8}{3} \frac{d}{d x} [x^{3}]$ .

$\frac{8}{3} \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 7 x]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$\frac{8}{3} (3 x^{2}) + \frac{d}{d x} [- 7 x]$

Step 2.3

Combine $3$ and $\frac{8}{3}$ .

$\frac{3 \cdot 8}{3} x^{2} + \frac{d}{d x} [- 7 x]$

Step 2.4

Multiply $3$ by $8$ .

$\frac{24}{3} x^{2} + \frac{d}{d x} [- 7 x]$

Step 2.5

Combine $\frac{24}{3}$ and $x^{2}$ .

$\frac{24 x^{2}}{3} + \frac{d}{d x} [- 7 x]$

Step 2.6

Cancel the common factor of $24$ and $3$ .

Tap for more steps...

Step 2.6.1

Factor $3$ out of $24 x^{2}$ .

$\frac{3 (8 x^{2})}{3} + \frac{d}{d x} [- 7 x]$

Step 2.6.2

Cancel the common factors.

Tap for more steps...

Step 2.6.2.1

Factor $3$ out of $3$ .

$\frac{3 (8 x^{2})}{3 (1)} + \frac{d}{d x} [- 7 x]$

Step 2.6.2.2

Cancel the common factor.

$\frac{3 (8 x^{2})}{3 \cdot 1} + \frac{d}{d x} [- 7 x]$

Step 2.6.2.3

Rewrite the expression.

$\frac{8 x^{2}}{1} + \frac{d}{d x} [- 7 x]$

Step 2.6.2.4

Divide $8 x^{2}$ by $1$ .

$8 x^{2} + \frac{d}{d x} [- 7 x]$

Step 3

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

Tap for more steps...

Step 3.1

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

$8 x^{2} - 7 \frac{d}{d x} [x]$

Step 3.2

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

$8 x^{2} - 7 \cdot 1$

Step 3.3

Multiply $- 7$ by $1$ .

$8 x^{2} - 7$