Calculus Examples

$y = \frac{x^{5}}{120}$

Step 1

Find the first derivative.

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Step 1.1

Since $\frac{1}{120}$ is constant with respect to $x$ , the derivative of $\frac{x^{5}}{120}$ with respect to $x$ is $\frac{1}{120} \frac{d}{d x} [x^{5}]$ .

$\frac{1}{120} \frac{d}{d x} [x^{5}]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$\frac{1}{120} (5 x^{4})$

Step 1.3

Simplify terms.

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Step 1.3.1

Combine $5$ and $\frac{1}{120}$ .

$\frac{5}{120} x^{4}$

Step 1.3.2

Combine $\frac{5}{120}$ and $x^{4}$ .

$\frac{5 x^{4}}{120}$

Step 1.3.3

Cancel the common factor of $5$ and $120$ .

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Step 1.3.3.1

Factor $5$ out of $5 x^{4}$ .

$\frac{5 (x^{4})}{120}$

Step 1.3.3.2

Cancel the common factors.

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Step 1.3.3.2.1

Factor $5$ out of $120$ .

$\frac{5 x^{4}}{5 \cdot 24}$

Step 1.3.3.2.2

Cancel the common factor.

$\frac{5 x^{4}}{5 \cdot 24}$

Step 1.3.3.2.3

Rewrite the expression.

$f' (x) = \frac{x^{4}}{24}$

Step 2

Find the second derivative.

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Step 2.1

Since $\frac{1}{24}$ is constant with respect to $x$ , the derivative of $\frac{x^{4}}{24}$ with respect to $x$ is $\frac{1}{24} \frac{d}{d x} [x^{4}]$ .

$\frac{1}{24} \frac{d}{d x} [x^{4}]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$\frac{1}{24} (4 x^{3})$

Step 2.3

Simplify terms.

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Step 2.3.1

Combine $4$ and $\frac{1}{24}$ .

$\frac{4}{24} x^{3}$

Step 2.3.2

Combine $\frac{4}{24}$ and $x^{3}$ .

$\frac{4 x^{3}}{24}$

Step 2.3.3

Cancel the common factor of $4$ and $24$ .

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Step 2.3.3.1

Factor $4$ out of $4 x^{3}$ .

$\frac{4 (x^{3})}{24}$

Step 2.3.3.2

Cancel the common factors.

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Step 2.3.3.2.1

Factor $4$ out of $24$ .

$\frac{4 x^{3}}{4 \cdot 6}$

Step 2.3.3.2.2

Cancel the common factor.

$\frac{4 x^{3}}{4 \cdot 6}$

Step 2.3.3.2.3

Rewrite the expression.

$f'' (x) = \frac{x^{3}}{6}$