Calculus Examples

$y = - \frac{5}{3} x^{4} + \frac{1}{4} x^{5} + \frac{4}{15}$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $- \frac{5}{3} x^{4} + \frac{1}{4} x^{5} + \frac{4}{15}$ with respect to $x$ is $\frac{d}{d x} [- \frac{5}{3} x^{4}] + \frac{d}{d x} [\frac{1}{4} x^{5}] + \frac{d}{d x} [\frac{4}{15}]$ .

$\frac{d}{d x} [- \frac{5}{3} x^{4}] + \frac{d}{d x} [\frac{1}{4} x^{5}] + \frac{d}{d x} [\frac{4}{15}]$

Step 1.2

Evaluate $\frac{d}{d x} [- \frac{5}{3} x^{4}]$ .

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Step 1.2.1

Since $- \frac{5}{3}$ is constant with respect to $x$ , the derivative of $- \frac{5}{3} x^{4}$ with respect to $x$ is $- \frac{5}{3} \frac{d}{d x} [x^{4}]$ .

$- \frac{5}{3} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [\frac{1}{4} x^{5}] + \frac{d}{d x} [\frac{4}{15}]$

Step 1.2.2

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

$- \frac{5}{3} (4 x^{3}) + \frac{d}{d x} [\frac{1}{4} x^{5}] + \frac{d}{d x} [\frac{4}{15}]$

Step 1.2.3

Multiply $4$ by $- 1$ .

$- 4 (\frac{5}{3}) x^{3} + \frac{d}{d x} [\frac{1}{4} x^{5}] + \frac{d}{d x} [\frac{4}{15}]$

Step 1.2.4

Combine $- 4$ and $\frac{5}{3}$ .

$\frac{- 4 \cdot 5}{3} x^{3} + \frac{d}{d x} [\frac{1}{4} x^{5}] + \frac{d}{d x} [\frac{4}{15}]$

Step 1.2.5

Multiply $- 4$ by $5$ .

$\frac{- 20}{3} x^{3} + \frac{d}{d x} [\frac{1}{4} x^{5}] + \frac{d}{d x} [\frac{4}{15}]$

Step 1.2.6

Combine $\frac{- 20}{3}$ and $x^{3}$ .

$\frac{- 20 x^{3}}{3} + \frac{d}{d x} [\frac{1}{4} x^{5}] + \frac{d}{d x} [\frac{4}{15}]$

Step 1.2.7

Move the negative in front of the fraction.

$- \frac{20 x^{3}}{3} + \frac{d}{d x} [\frac{1}{4} x^{5}] + \frac{d}{d x} [\frac{4}{15}]$

Step 1.3

Evaluate $\frac{d}{d x} [\frac{1}{4} x^{5}]$ .

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Step 1.3.1

Since $\frac{1}{4}$ is constant with respect to $x$ , the derivative of $\frac{1}{4} x^{5}$ with respect to $x$ is $\frac{1}{4} \frac{d}{d x} [x^{5}]$ .

$- \frac{20 x^{3}}{3} + \frac{1}{4} \frac{d}{d x} [x^{5}] + \frac{d}{d x} [\frac{4}{15}]$

Step 1.3.2

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

$- \frac{20 x^{3}}{3} + \frac{1}{4} (5 x^{4}) + \frac{d}{d x} [\frac{4}{15}]$

Step 1.3.3

Combine $5$ and $\frac{1}{4}$ .

$- \frac{20 x^{3}}{3} + \frac{5}{4} x^{4} + \frac{d}{d x} [\frac{4}{15}]$

Step 1.3.4

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

$- \frac{20 x^{3}}{3} + \frac{5 x^{4}}{4} + \frac{d}{d x} [\frac{4}{15}]$

Step 1.4

Since $\frac{4}{15}$ is constant with respect to $x$ , the derivative of $\frac{4}{15}$ with respect to $x$ is $0$ .

$- \frac{20 x^{3}}{3} + \frac{5 x^{4}}{4} + 0$

Step 1.5

Simplify.

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Step 1.5.1

Add $- \frac{20 x^{3}}{3} + \frac{5 x^{4}}{4}$ and $0$ .

$- \frac{20 x^{3}}{3} + \frac{5 x^{4}}{4}$

Step 1.5.2

Reorder terms.

$f' (x) = \frac{5 x^{4}}{4} - \frac{20 x^{3}}{3}$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $\frac{5 x^{4}}{4} - \frac{20 x^{3}}{3}$ with respect to $x$ is $\frac{d}{d x} [\frac{5 x^{4}}{4}] + \frac{d}{d x} [- \frac{20 x^{3}}{3}]$ .

$\frac{d}{d x} [\frac{5 x^{4}}{4}] + \frac{d}{d x} [- \frac{20 x^{3}}{3}]$

Step 2.2

Evaluate $\frac{d}{d x} [\frac{5 x^{4}}{4}]$ .

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Step 2.2.1

Since $\frac{5}{4}$ is constant with respect to $x$ , the derivative of $\frac{5 x^{4}}{4}$ with respect to $x$ is $\frac{5}{4} \frac{d}{d x} [x^{4}]$ .

$\frac{5}{4} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- \frac{20 x^{3}}{3}]$

Step 2.2.2

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

$\frac{5}{4} (4 x^{3}) + \frac{d}{d x} [- \frac{20 x^{3}}{3}]$

Step 2.2.3

Combine $4$ and $\frac{5}{4}$ .

$\frac{4 \cdot 5}{4} x^{3} + \frac{d}{d x} [- \frac{20 x^{3}}{3}]$

Step 2.2.4

Multiply $4$ by $5$ .

$\frac{20}{4} x^{3} + \frac{d}{d x} [- \frac{20 x^{3}}{3}]$

Step 2.2.5

Combine $\frac{20}{4}$ and $x^{3}$ .

$\frac{20 x^{3}}{4} + \frac{d}{d x} [- \frac{20 x^{3}}{3}]$

Step 2.2.6

Cancel the common factor of $20$ and $4$ .

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Step 2.2.6.1

Factor $4$ out of $20 x^{3}$ .

$\frac{4 (5 x^{3})}{4} + \frac{d}{d x} [- \frac{20 x^{3}}{3}]$

Step 2.2.6.2

Cancel the common factors.

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Step 2.2.6.2.1

Factor $4$ out of $4$ .

$\frac{4 (5 x^{3})}{4 (1)} + \frac{d}{d x} [- \frac{20 x^{3}}{3}]$

Step 2.2.6.2.2

Cancel the common factor.

$\frac{4 (5 x^{3})}{4 \cdot 1} + \frac{d}{d x} [- \frac{20 x^{3}}{3}]$

Step 2.2.6.2.3

Rewrite the expression.

$\frac{5 x^{3}}{1} + \frac{d}{d x} [- \frac{20 x^{3}}{3}]$

Step 2.2.6.2.4

Divide $5 x^{3}$ by $1$ .

$5 x^{3} + \frac{d}{d x} [- \frac{20 x^{3}}{3}]$

Step 2.3

Evaluate $\frac{d}{d x} [- \frac{20 x^{3}}{3}]$ .

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Step 2.3.1

Since $- \frac{20}{3}$ is constant with respect to $x$ , the derivative of $- \frac{20 x^{3}}{3}$ with respect to $x$ is $- \frac{20}{3} \frac{d}{d x} [x^{3}]$ .

$5 x^{3} - \frac{20}{3} \frac{d}{d x} [x^{3}]$

Step 2.3.2

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

$5 x^{3} - \frac{20}{3} (3 x^{2})$

Step 2.3.3

Multiply $3$ by $- 1$ .

$5 x^{3} - 3 (\frac{20}{3}) x^{2}$

Step 2.3.4

Combine $- 3$ and $\frac{20}{3}$ .

$5 x^{3} + \frac{- 3 \cdot 20}{3} x^{2}$

Step 2.3.5

Multiply $- 3$ by $20$ .

$5 x^{3} + \frac{- 60}{3} x^{2}$

Step 2.3.6

Combine $\frac{- 60}{3}$ and $x^{2}$ .

$5 x^{3} + \frac{- 60 x^{2}}{3}$

Step 2.3.7

Cancel the common factor of $- 60$ and $3$ .

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Step 2.3.7.1

Factor $3$ out of $- 60 x^{2}$ .

$5 x^{3} + \frac{3 (- 20 x^{2})}{3}$

Step 2.3.7.2

Cancel the common factors.

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Step 2.3.7.2.1

Factor $3$ out of $3$ .

$5 x^{3} + \frac{3 (- 20 x^{2})}{3 (1)}$

Step 2.3.7.2.2

Cancel the common factor.

$5 x^{3} + \frac{3 (- 20 x^{2})}{3 \cdot 1}$

Step 2.3.7.2.3

Rewrite the expression.

$5 x^{3} + \frac{- 20 x^{2}}{1}$

Step 2.3.7.2.4

Divide $- 20 x^{2}$ by $1$ .

$f'' (x) = 5 x^{3} - 20 x^{2}$