Calculus Examples

$y = \frac{x^{3}}{3}$ solve $y'''' = x^{2}$

Step 1

Find $y'$ .

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Step 1.1

Differentiate both sides of the equation.

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

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

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

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

Step 1.3.2

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

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

Step 1.3.3

Simplify terms.

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Step 1.3.3.1

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

$\frac{3}{3} x^{2}$

Step 1.3.3.2

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

$\frac{3 x^{2}}{3}$

Step 1.3.3.3

Cancel the common factor of $3$ .

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Step 1.3.3.3.1

Cancel the common factor.

$\frac{3 x^{2}}{3}$

Step 1.3.3.3.2

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

$x^{2}$

Step 1.4

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

$y' = x^{2}$

Step 2

Find $y''$ .

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Step 2.1

Set up the derivative.

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

Step 2.2

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

$y'' = 2 x$

Step 3

Find $y'''$ .

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Step 3.1

Set up the derivative.

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

Step 3.2

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

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

Step 3.3

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

$y''' = 2 \cdot 1$

Step 3.4

Multiply $2$ by $1$ .

$y''' = 2$

Step 4

Find $y''''$ .

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Step 4.1

Set up the derivative.

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

Step 4.2

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

$y'''' = 0$

Step 5

Substitute into the given differential equation.

$0 = x^{2}$

Step 6

The given solution does not satisfy the given differential equation.

$y = \frac{x^{3}}{3}$ is not a solution to $y'''' = x^{2}$