Calculus Examples

$y = x - \frac{1}{90} x^{3}$

Step 1

Differentiate.

Tap for more steps...

Step 1.1

By the Sum Rule, the derivative of $x - \frac{1}{90} x^{3}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{1}{90} x^{3}]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{1}{90} x^{3}]$

Step 1.2

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

$1 + \frac{d}{d x} [- \frac{1}{90} x^{3}]$

Step 2

Evaluate $\frac{d}{d x} [- \frac{1}{90} x^{3}]$ .

Tap for more steps...

Step 2.1

Since $- \frac{1}{90}$ is constant with respect to $x$ , the derivative of $- \frac{1}{90} x^{3}$ with respect to $x$ is $- \frac{1}{90} \frac{d}{d x} [x^{3}]$ .

$1 - \frac{1}{90} \frac{d}{d x} [x^{3}]$

Step 2.2

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

$1 - \frac{1}{90} (3 x^{2})$

Step 2.3

Multiply $3$ by $- 1$ .

$1 - 3 (\frac{1}{90}) x^{2}$

Step 2.4

Combine $- 3$ and $\frac{1}{90}$ .

$1 + \frac{- 3}{90} x^{2}$

Step 2.5

Combine $\frac{- 3}{90}$ and $x^{2}$ .

$1 + \frac{- 3 x^{2}}{90}$

Step 2.6

Cancel the common factor of $- 3$ and $90$ .

Tap for more steps...

Step 2.6.1

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

$1 + \frac{3 (- x^{2})}{90}$

Step 2.6.2

Cancel the common factors.

Tap for more steps...

Step 2.6.2.1

Factor $3$ out of $90$ .

$1 + \frac{3 (- x^{2})}{3 (30)}$

Step 2.6.2.2

Cancel the common factor.

$1 + \frac{3 (- x^{2})}{3 \cdot 30}$

Step 2.6.2.3

Rewrite the expression.

$1 + \frac{- x^{2}}{30}$

Step 2.7

Move the negative in front of the fraction.

$1 - \frac{x^{2}}{30}$

Step 3

Reorder terms.

$- \frac{x^{2}}{30} + 1$