Calculus Examples

$y' = 3 x^{2}$ , $y = x^{3} - 4$

Step 1

Find $y'$ .

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Step 1.1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (x^{3} - 4)$

Step 1.2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 1.3

Differentiate the right side of the equation.

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Step 1.3.1

By the Sum Rule, the derivative of $x^{3} - 4$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 4]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 4]$

Step 1.3.2

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

$3 x^{2} + \frac{d}{d x} [- 4]$

Step 1.3.3

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$3 x^{2} + 0$

Step 1.3.4

Add $3 x^{2}$ and $0$ .

$3 x^{2}$

Step 1.4

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

$y' = 3 x^{2}$

Step 2

Substitute into the given differential equation.

$3 x^{2} = 3 x^{2}$

Step 3

The given solution satisfies the given differential equation.

$y = x^{3} - 4$ is a solution to $y' = 3 x^{2}$