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

Find $y''$ .

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Step 2.1

Set up the derivative.

$y'' = \frac{d}{d x} [3 x^{2}]$

Step 2.2

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

$y'' = 3 \frac{d}{d x} [x^{2}]$

Step 2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$y'' = 3 (2 x)$

Step 2.4

Multiply $2$ by $3$ .

$y'' = 6 x$

Step 3

Find $y'''$ .

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Step 3.1

Set up the derivative.

$y''' = \frac{d}{d x} [6 x]$

Step 3.2

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$y''' = 6 \frac{d}{d x} [x]$

Step 3.3

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

$y''' = 6 \cdot 1$

Step 3.4

Multiply $6$ by $1$ .

$y''' = 6$

Step 4

Find $y''''$ .

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Step 4.1

Set up the derivative.

$y'''' = \frac{d}{d x} [6]$

Step 4.2

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$y'''' = 0$

Step 5

Substitute into the given differential equation.

$0 = 3 x^{2}$

Step 6

The given solution does not satisfy the given differential equation.

$y = x^{3} - 4$ is not a solution to $y'''' = 3 x^{2}$