Calculus Examples

$y = - \frac{1}{5} x^{- 4} + \frac{1}{2} x^{3} - 3 x^{- 1}$

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

By the Sum Rule, the derivative of $- \frac{1}{5} x^{- 4} + \frac{1}{2} x^{3} - 3 x^{- 1}$ with respect to $x$ is $\frac{d}{d x} [- \frac{1}{5} x^{- 4}] + \frac{d}{d x} [\frac{1}{2} x^{3}] + \frac{d}{d x} [- 3 x^{- 1}]$ .

$\frac{d}{d x} [- \frac{1}{5} x^{- 4}] + \frac{d}{d x} [\frac{1}{2} x^{3}] + \frac{d}{d x} [- 3 x^{- 1}]$

Step 1.2

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

Tap for more steps...

Step 1.2.1

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

$- \frac{1}{5} \frac{d}{d x} [x^{- 4}] + \frac{d}{d x} [\frac{1}{2} x^{3}] + \frac{d}{d x} [- 3 x^{- 1}]$

Step 1.2.2

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

$- \frac{1}{5} (- 4 x^{- 5}) + \frac{d}{d x} [\frac{1}{2} x^{3}] + \frac{d}{d x} [- 3 x^{- 1}]$

Step 1.2.3

Multiply $- 4$ by $- 1$ .

$4 (\frac{1}{5}) x^{- 5} + \frac{d}{d x} [\frac{1}{2} x^{3}] + \frac{d}{d x} [- 3 x^{- 1}]$

Step 1.2.4

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

$\frac{4}{5} x^{- 5} + \frac{d}{d x} [\frac{1}{2} x^{3}] + \frac{d}{d x} [- 3 x^{- 1}]$

Step 1.2.5

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

$\frac{4 x^{- 5}}{5} + \frac{d}{d x} [\frac{1}{2} x^{3}] + \frac{d}{d x} [- 3 x^{- 1}]$

Step 1.2.6

Move $x^{- 5}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{4}{5 x^{5}} + \frac{d}{d x} [\frac{1}{2} x^{3}] + \frac{d}{d x} [- 3 x^{- 1}]$

Step 1.3

Evaluate $\frac{d}{d x} [\frac{1}{2} x^{3}]$ .

Tap for more steps...

Step 1.3.1

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{1}{2} x^{3}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x^{3}]$ .

$\frac{4}{5 x^{5}} + \frac{1}{2} \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 3 x^{- 1}]$

Step 1.3.2

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

$\frac{4}{5 x^{5}} + \frac{1}{2} (3 x^{2}) + \frac{d}{d x} [- 3 x^{- 1}]$

Step 1.3.3

Combine $3$ and $\frac{1}{2}$ .

$\frac{4}{5 x^{5}} + \frac{3}{2} x^{2} + \frac{d}{d x} [- 3 x^{- 1}]$

Step 1.3.4

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

$\frac{4}{5 x^{5}} + \frac{3 x^{2}}{2} + \frac{d}{d x} [- 3 x^{- 1}]$

Step 1.4

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

Tap for more steps...

Step 1.4.1

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

$\frac{4}{5 x^{5}} + \frac{3 x^{2}}{2} - 3 \frac{d}{d x} [x^{- 1}]$

Step 1.4.2

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

$\frac{4}{5 x^{5}} + \frac{3 x^{2}}{2} - 3 (- x^{- 2})$

Step 1.4.3

Multiply $- 1$ by $- 3$ .

$\frac{4}{5 x^{5}} + \frac{3 x^{2}}{2} + 3 x^{- 2}$

Step 1.5

Simplify.

Tap for more steps...

Step 1.5.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{4}{5 x^{5}} + \frac{3 x^{2}}{2} + 3 \frac{1}{x^{2}}$

Step 1.5.2

Combine $3$ and $\frac{1}{x^{2}}$ .

$\frac{4}{5 x^{5}} + \frac{3 x^{2}}{2} + \frac{3}{x^{2}}$

Step 1.5.3

Reorder terms.

$f' (x) = \frac{3 x^{2}}{2} + \frac{3}{x^{2}} + \frac{4}{5 x^{5}}$

Step 2

Find the second derivative.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $\frac{3 x^{2}}{2} + \frac{3}{x^{2}} + \frac{4}{5 x^{5}}$ with respect to $x$ is $\frac{d}{d x} [\frac{3 x^{2}}{2}] + \frac{d}{d x} [\frac{3}{x^{2}}] + \frac{d}{d x} [\frac{4}{5 x^{5}}]$ .

$\frac{d}{d x} [\frac{3 x^{2}}{2}] + \frac{d}{d x} [\frac{3}{x^{2}}] + \frac{d}{d x} [\frac{4}{5 x^{5}}]$

Step 2.2

Evaluate $\frac{d}{d x} [\frac{3 x^{2}}{2}]$ .

Tap for more steps...

Step 2.2.1

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

$\frac{3}{2} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [\frac{3}{x^{2}}] + \frac{d}{d x} [\frac{4}{5 x^{5}}]$

Step 2.2.2

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

$\frac{3}{2} (2 x) + \frac{d}{d x} [\frac{3}{x^{2}}] + \frac{d}{d x} [\frac{4}{5 x^{5}}]$

Step 2.2.3

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

$\frac{2 \cdot 3}{2} x + \frac{d}{d x} [\frac{3}{x^{2}}] + \frac{d}{d x} [\frac{4}{5 x^{5}}]$

Step 2.2.4

Multiply $2$ by $3$ .

$\frac{6}{2} x + \frac{d}{d x} [\frac{3}{x^{2}}] + \frac{d}{d x} [\frac{4}{5 x^{5}}]$

Step 2.2.5

Combine $\frac{6}{2}$ and $x$ .

$\frac{6 x}{2} + \frac{d}{d x} [\frac{3}{x^{2}}] + \frac{d}{d x} [\frac{4}{5 x^{5}}]$

Step 2.2.6

Cancel the common factor of $6$ and $2$ .

Tap for more steps...

Step 2.2.6.1

Factor $2$ out of $6 x$ .

$\frac{2 (3 x)}{2} + \frac{d}{d x} [\frac{3}{x^{2}}] + \frac{d}{d x} [\frac{4}{5 x^{5}}]$

Step 2.2.6.2

Cancel the common factors.

Tap for more steps...

Step 2.2.6.2.1

Factor $2$ out of $2$ .

$\frac{2 (3 x)}{2 (1)} + \frac{d}{d x} [\frac{3}{x^{2}}] + \frac{d}{d x} [\frac{4}{5 x^{5}}]$

Step 2.2.6.2.2

Cancel the common factor.

$\frac{2 (3 x)}{2 \cdot 1} + \frac{d}{d x} [\frac{3}{x^{2}}] + \frac{d}{d x} [\frac{4}{5 x^{5}}]$

Step 2.2.6.2.3

Rewrite the expression.

$\frac{3 x}{1} + \frac{d}{d x} [\frac{3}{x^{2}}] + \frac{d}{d x} [\frac{4}{5 x^{5}}]$

Step 2.2.6.2.4

Divide $3 x$ by $1$ .

$3 x + \frac{d}{d x} [\frac{3}{x^{2}}] + \frac{d}{d x} [\frac{4}{5 x^{5}}]$

Step 2.3

Evaluate $\frac{d}{d x} [\frac{3}{x^{2}}]$ .

Tap for more steps...

Step 2.3.1

Since $3$ is constant with respect to $x$ , the derivative of $\frac{3}{x^{2}}$ with respect to $x$ is $3 \frac{d}{d x} [\frac{1}{x^{2}}]$ .

$3 x + 3 \frac{d}{d x} [\frac{1}{x^{2}}] + \frac{d}{d x} [\frac{4}{5 x^{5}}]$

Step 2.3.2

Rewrite $\frac{1}{x^{2}}$ as ${(x^{2})}^{- 1}$ .

$3 x + 3 \frac{d}{d x} [{(x^{2})}^{- 1}] + \frac{d}{d x} [\frac{4}{5 x^{5}}]$

Step 2.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{2}$ .

Tap for more steps...

Step 2.3.3.1

To apply the Chain Rule, set $u_{1}$ as $x^{2}$ .

$3 x + 3 (\frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [\frac{4}{5 x^{5}}]$

Step 2.3.3.2

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

$3 x + 3 (- {u_{1}}^{- 2} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [\frac{4}{5 x^{5}}]$

Step 2.3.3.3

Replace all occurrences of $u_{1}$ with $x^{2}$ .

$3 x + 3 (- {(x^{2})}^{- 2} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [\frac{4}{5 x^{5}}]$

Step 2.3.4

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

$3 x + 3 (- {(x^{2})}^{- 2} (2 x)) + \frac{d}{d x} [\frac{4}{5 x^{5}}]$

Step 2.3.5

Multiply the exponents in ${(x^{2})}^{- 2}$ .

Tap for more steps...

Step 2.3.5.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$3 x + 3 (- x^{2 \cdot - 2} (2 x)) + \frac{d}{d x} [\frac{4}{5 x^{5}}]$

Step 2.3.5.2

Multiply $2$ by $- 2$ .

$3 x + 3 (- x^{- 4} (2 x)) + \frac{d}{d x} [\frac{4}{5 x^{5}}]$

Step 2.3.6

Multiply $2$ by $- 1$ .

$3 x + 3 (- 2 x^{- 4} x) + \frac{d}{d x} [\frac{4}{5 x^{5}}]$

Step 2.3.7

Raise $x$ to the power of $1$ .

$3 x + 3 (- 2 (x^{1} x^{- 4})) + \frac{d}{d x} [\frac{4}{5 x^{5}}]$

Step 2.3.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$3 x + 3 (- 2 x^{1 - 4}) + \frac{d}{d x} [\frac{4}{5 x^{5}}]$

Step 2.3.9

Subtract $4$ from $1$ .

$3 x + 3 (- 2 x^{- 3}) + \frac{d}{d x} [\frac{4}{5 x^{5}}]$

Step 2.3.10

Multiply $- 2$ by $3$ .

$3 x - 6 x^{- 3} + \frac{d}{d x} [\frac{4}{5 x^{5}}]$

Step 2.4

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

Tap for more steps...

Step 2.4.1

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

$3 x - 6 x^{- 3} + \frac{4}{5} \frac{d}{d x} [\frac{1}{x^{5}}]$

Step 2.4.2

Rewrite $\frac{1}{x^{5}}$ as ${(x^{5})}^{- 1}$ .

$3 x - 6 x^{- 3} + \frac{4}{5} \frac{d}{d x} [{(x^{5})}^{- 1}]$

Step 2.4.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{5}$ .

Tap for more steps...

Step 2.4.3.1

To apply the Chain Rule, set $u_{2}$ as $x^{5}$ .

$3 x - 6 x^{- 3} + \frac{4}{5} (\frac{d}{d u_{2}} [{u_{2}}^{- 1}] \frac{d}{d x} [x^{5}])$

Step 2.4.3.2

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

$3 x - 6 x^{- 3} + \frac{4}{5} (- {u_{2}}^{- 2} \frac{d}{d x} [x^{5}])$

Step 2.4.3.3

Replace all occurrences of $u_{2}$ with $x^{5}$ .

$3 x - 6 x^{- 3} + \frac{4}{5} (- {(x^{5})}^{- 2} \frac{d}{d x} [x^{5}])$

Step 2.4.4

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

$3 x - 6 x^{- 3} + \frac{4}{5} (- {(x^{5})}^{- 2} (5 x^{4}))$

Step 2.4.5

Multiply the exponents in ${(x^{5})}^{- 2}$ .

Tap for more steps...

Step 2.4.5.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$3 x - 6 x^{- 3} + \frac{4}{5} (- x^{5 \cdot - 2} (5 x^{4}))$

Step 2.4.5.2

Multiply $5$ by $- 2$ .

$3 x - 6 x^{- 3} + \frac{4}{5} (- x^{- 10} (5 x^{4}))$

Step 2.4.6

Multiply $5$ by $- 1$ .

$3 x - 6 x^{- 3} + \frac{4}{5} (- 5 x^{- 10} x^{4})$

Step 2.4.7

Multiply $x^{- 10}$ by $x^{4}$ by adding the exponents.

Tap for more steps...

Step 2.4.7.1

Move $x^{4}$ .

$3 x - 6 x^{- 3} + \frac{4}{5} (- 5 (x^{4} x^{- 10}))$

Step 2.4.7.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$3 x - 6 x^{- 3} + \frac{4}{5} (- 5 x^{4 - 10})$

Step 2.4.7.3

Subtract $10$ from $4$ .

$3 x - 6 x^{- 3} + \frac{4}{5} (- 5 x^{- 6})$

Step 2.4.8

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

$3 x - 6 x^{- 3} + \frac{- 5 \cdot 4}{5} x^{- 6}$

Step 2.4.9

Multiply $- 5$ by $4$ .

$3 x - 6 x^{- 3} + \frac{- 20}{5} x^{- 6}$

Step 2.4.10

Combine $\frac{- 20}{5}$ and $x^{- 6}$ .

$3 x - 6 x^{- 3} + \frac{- 20 x^{- 6}}{5}$

Step 2.4.11

Move $x^{- 6}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$3 x - 6 x^{- 3} + \frac{- 20}{5 x^{6}}$

Step 2.4.12

Cancel the common factor of $- 20$ and $5$ .

Tap for more steps...

Step 2.4.12.1

Factor $5$ out of $- 20$ .

$3 x - 6 x^{- 3} + \frac{5 \cdot - 4}{5 x^{6}}$

Step 2.4.12.2

Cancel the common factors.

Tap for more steps...

Step 2.4.12.2.1

Factor $5$ out of $5 x^{6}$ .

$3 x - 6 x^{- 3} + \frac{5 \cdot - 4}{5 (x^{6})}$

Step 2.4.12.2.2

Cancel the common factor.

$3 x - 6 x^{- 3} + \frac{5 \cdot - 4}{5 x^{6}}$

Step 2.4.12.2.3

Rewrite the expression.

$3 x - 6 x^{- 3} + \frac{- 4}{x^{6}}$

Step 2.4.13

Move the negative in front of the fraction.

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

Step 2.5

Simplify.

Tap for more steps...

Step 2.5.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$3 x - 6 \frac{1}{x^{3}} - \frac{4}{x^{6}}$

Step 2.5.2

Combine terms.

Tap for more steps...

Step 2.5.2.1

Combine $- 6$ and $\frac{1}{x^{3}}$ .

$3 x + \frac{- 6}{x^{3}} - \frac{4}{x^{6}}$

Step 2.5.2.2

Move the negative in front of the fraction.

$f'' (x) = 3 x - \frac{6}{x^{3}} - \frac{4}{x^{6}}$