Calculus Examples

$f (x) = 2 x^{3.1} - \frac{4}{x^{5}} + 4 \sqrt{x^{5}} - x$

Step 1

By the Sum Rule, the derivative of $2 x^{3.1} - \frac{4}{x^{5}} + 4 \sqrt{x^{5}} - x$ with respect to $x$ is $\frac{d}{d x} [2 x^{3.1}] + \frac{d}{d x} [- \frac{4}{x^{5}}] + \frac{d}{d x} [4 \sqrt{x^{5}}] + \frac{d}{d x} [- x]$ .

$\frac{d}{d x} [2 x^{3.1}] + \frac{d}{d x} [- \frac{4}{x^{5}}] + \frac{d}{d x} [4 \sqrt{x^{5}}] + \frac{d}{d x} [- x]$

Step 2

Evaluate $\frac{d}{d x} [2 x^{3.1}]$ .

Tap for more steps...

Step 2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{3.1}$ with respect to $x$ is $2 \frac{d}{d x} [x^{3.1}]$ .

$2 \frac{d}{d x} [x^{3.1}] + \frac{d}{d x} [- \frac{4}{x^{5}}] + \frac{d}{d x} [4 \sqrt{x^{5}}] + \frac{d}{d x} [- x]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3.1$ .

$2 (3.1 x^{2.1}) + \frac{d}{d x} [- \frac{4}{x^{5}}] + \frac{d}{d x} [4 \sqrt{x^{5}}] + \frac{d}{d x} [- x]$

Step 2.3

Multiply $3.1$ by $2$ .

$6.2 x^{2.1} + \frac{d}{d x} [- \frac{4}{x^{5}}] + \frac{d}{d x} [4 \sqrt{x^{5}}] + \frac{d}{d x} [- x]$

Step 3

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

Tap for more steps...

Step 3.1

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

$6.2 x^{2.1} - 4 \frac{d}{d x} [\frac{1}{x^{5}}] + \frac{d}{d x} [4 \sqrt{x^{5}}] + \frac{d}{d x} [- x]$

Step 3.2

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

$6.2 x^{2.1} - 4 \frac{d}{d x} [{(x^{5})}^{- 1}] + \frac{d}{d x} [4 \sqrt{x^{5}}] + \frac{d}{d x} [- x]$

Step 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 $g (x) = x^{5}$ .

Tap for more steps...

Step 3.3.1

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

$6.2 x^{2.1} - 4 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{5}]) + \frac{d}{d x} [4 \sqrt{x^{5}}] + \frac{d}{d x} [- x]$

Step 3.3.2

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

$6.2 x^{2.1} - 4 (- u^{- 2} \frac{d}{d x} [x^{5}]) + \frac{d}{d x} [4 \sqrt{x^{5}}] + \frac{d}{d x} [- x]$

Step 3.3.3

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

$6.2 x^{2.1} - 4 (- {(x^{5})}^{- 2} \frac{d}{d x} [x^{5}]) + \frac{d}{d x} [4 \sqrt{x^{5}}] + \frac{d}{d x} [- x]$

Step 3.4

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

$6.2 x^{2.1} - 4 (- {(x^{5})}^{- 2} (5 x^{4})) + \frac{d}{d x} [4 \sqrt{x^{5}}] + \frac{d}{d x} [- x]$

Step 3.5

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

Tap for more steps...

Step 3.5.1

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

$6.2 x^{2.1} - 4 (- x^{5 \cdot - 2} (5 x^{4})) + \frac{d}{d x} [4 \sqrt{x^{5}}] + \frac{d}{d x} [- x]$

Step 3.5.2

Multiply $5$ by $- 2$ .

$6.2 x^{2.1} - 4 (- x^{- 10} (5 x^{4})) + \frac{d}{d x} [4 \sqrt{x^{5}}] + \frac{d}{d x} [- x]$

Step 3.6

Multiply $5$ by $- 1$ .

$6.2 x^{2.1} - 4 (- 5 x^{- 10} x^{4}) + \frac{d}{d x} [4 \sqrt{x^{5}}] + \frac{d}{d x} [- x]$

Step 3.7

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

Tap for more steps...

Step 3.7.1

Move $x^{4}$ .

$6.2 x^{2.1} - 4 (- 5 (x^{4} x^{- 10})) + \frac{d}{d x} [4 \sqrt{x^{5}}] + \frac{d}{d x} [- x]$

Step 3.7.2

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

$6.2 x^{2.1} - 4 (- 5 x^{4 - 10}) + \frac{d}{d x} [4 \sqrt{x^{5}}] + \frac{d}{d x} [- x]$

Step 3.7.3

Subtract $10$ from $4$ .

$6.2 x^{2.1} - 4 (- 5 x^{- 6}) + \frac{d}{d x} [4 \sqrt{x^{5}}] + \frac{d}{d x} [- x]$

Step 3.8

Multiply $- 5$ by $- 4$ .

$6.2 x^{2.1} + 20 x^{- 6} + \frac{d}{d x} [4 \sqrt{x^{5}}] + \frac{d}{d x} [- x]$

Step 4

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

Tap for more steps...

Step 4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{5}}$ as $x^{\frac{5}{2}}$ .

$6.2 x^{2.1} + 20 x^{- 6} + \frac{d}{d x} [4 x^{\frac{5}{2}}] + \frac{d}{d x} [- x]$

Step 4.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{\frac{5}{2}}$ with respect to $x$ is $4 \frac{d}{d x} [x^{\frac{5}{2}}]$ .

$6.2 x^{2.1} + 20 x^{- 6} + 4 \frac{d}{d x} [x^{\frac{5}{2}}] + \frac{d}{d x} [- x]$

Step 4.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{5}{2}$ .

$6.2 x^{2.1} + 20 x^{- 6} + 4 (\frac{5}{2} x^{\frac{5}{2} - 1}) + \frac{d}{d x} [- x]$

Step 4.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$6.2 x^{2.1} + 20 x^{- 6} + 4 (\frac{5}{2} x^{\frac{5}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [- x]$

Step 4.5

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

$6.2 x^{2.1} + 20 x^{- 6} + 4 (\frac{5}{2} x^{\frac{5}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [- x]$

Step 4.6

Combine the numerators over the common denominator.

$6.2 x^{2.1} + 20 x^{- 6} + 4 (\frac{5}{2} x^{\frac{5 - 1 \cdot 2}{2}}) + \frac{d}{d x} [- x]$

Step 4.7

Simplify the numerator.

Tap for more steps...

Step 4.7.1

Multiply $- 1$ by $2$ .

$6.2 x^{2.1} + 20 x^{- 6} + 4 (\frac{5}{2} x^{\frac{5 - 2}{2}}) + \frac{d}{d x} [- x]$

Step 4.7.2

Subtract $2$ from $5$ .

$6.2 x^{2.1} + 20 x^{- 6} + 4 (\frac{5}{2} x^{\frac{3}{2}}) + \frac{d}{d x} [- x]$

Step 4.8

Combine $\frac{5}{2}$ and $x^{\frac{3}{2}}$ .

$6.2 x^{2.1} + 20 x^{- 6} + 4 \frac{5 x^{\frac{3}{2}}}{2} + \frac{d}{d x} [- x]$

Step 4.9

Combine $4$ and $\frac{5 x^{\frac{3}{2}}}{2}$ .

$6.2 x^{2.1} + 20 x^{- 6} + \frac{4 (5 x^{\frac{3}{2}})}{2} + \frac{d}{d x} [- x]$

Step 4.10

Multiply $5$ by $4$ .

$6.2 x^{2.1} + 20 x^{- 6} + \frac{20 x^{\frac{3}{2}}}{2} + \frac{d}{d x} [- x]$

Step 4.11

Factor $2$ out of $20 x^{\frac{3}{2}}$ .

$6.2 x^{2.1} + 20 x^{- 6} + \frac{2 (10 x^{\frac{3}{2}})}{2} + \frac{d}{d x} [- x]$

Step 4.12

Cancel the common factors.

Tap for more steps...

Step 4.12.1

Factor $2$ out of $2$ .

$6.2 x^{2.1} + 20 x^{- 6} + \frac{2 (10 x^{\frac{3}{2}})}{2 (1)} + \frac{d}{d x} [- x]$

Step 4.12.2

Cancel the common factor.

$6.2 x^{2.1} + 20 x^{- 6} + \frac{2 (10 x^{\frac{3}{2}})}{2 \cdot 1} + \frac{d}{d x} [- x]$

Step 4.12.3

Rewrite the expression.

$6.2 x^{2.1} + 20 x^{- 6} + \frac{10 x^{\frac{3}{2}}}{1} + \frac{d}{d x} [- x]$

Step 4.12.4

Divide $10 x^{\frac{3}{2}}$ by $1$ .

$6.2 x^{2.1} + 20 x^{- 6} + 10 x^{\frac{3}{2}} + \frac{d}{d x} [- x]$

Step 5

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

Tap for more steps...

Step 5.1

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

$6.2 x^{2.1} + 20 x^{- 6} + 10 x^{\frac{3}{2}} - \frac{d}{d x} [x]$

Step 5.2

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

$6.2 x^{2.1} + 20 x^{- 6} + 10 x^{\frac{3}{2}} - 1 \cdot 1$

Step 5.3

Multiply $- 1$ by $1$ .

$6.2 x^{2.1} + 20 x^{- 6} + 10 x^{\frac{3}{2}} - 1$

Step 6

Simplify.

Tap for more steps...

Step 6.1

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

$6.2 x^{2.1} + 20 \frac{1}{x^{6}} + 10 x^{\frac{3}{2}} - 1$

Step 6.2

Combine $20$ and $\frac{1}{x^{6}}$ .

$6.2 x^{2.1} + \frac{20}{x^{6}} + 10 x^{\frac{3}{2}} - 1$

Step 6.3

Reorder terms.

$\frac{20}{x^{6}} + 10 x^{\frac{3}{2}} + 6.2 x^{2.1} - 1$