Calculus Examples

$x + y - 2 x^{3} = 2$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (x + y - 2 x^{3}) = \frac{d}{d x} (2)$

Step 2

Differentiate the left side of the equation.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $x + y - 2 x^{3}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [y] + \frac{d}{d x} [- 2 x^{3}]$ .

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

Step 2.2

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

$1 + \frac{d}{d x} [y] + \frac{d}{d x} [- 2 x^{3}]$

Step 2.3

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$1 + y' + \frac{d}{d x} [- 2 x^{3}]$

Step 2.4

Evaluate $\frac{d}{d x} [- 2 x^{3}]$ .

Tap for more steps...

Step 2.4.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x^{3}$ with respect to $x$ is $- 2 \frac{d}{d x} [x^{3}]$ .

$1 + y' - 2 \frac{d}{d x} [x^{3}]$

Step 2.4.2

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

$1 + y' - 2 (3 x^{2})$

Step 2.4.3

Multiply $3$ by $- 2$ .

$1 + y' - 6 x^{2}$

Step 2.5

Reorder terms.

$- 6 x^{2} + 1 + y'$

Step 3

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$0$

Step 4

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

$- 6 x^{2} + 1 + y' = 0$

Step 5

Move all terms not containing $y'$ to the right side of the equation.

Tap for more steps...

Step 5.1

Add $6 x^{2}$ to both sides of the equation.

$1 + y' = 6 x^{2}$

Step 5.2

Subtract $1$ from both sides of the equation.

$y' = 6 x^{2} - 1$

Step 6

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

$\frac{d y}{d x} = 6 x^{2} - 1$