Calculus Examples

$y = \frac{x^{4}}{\sqrt{1 - 2 x}}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 - 2 x}$ as ${(1 - 2 x)}^{\frac{1}{2}}$ .

$y = \frac{x^{4}}{{(1 - 2 x)}^{\frac{1}{2}}}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{x^{4}}{{(1 - 2 x)}^{\frac{1}{2}}})$

Step 3

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 4

Differentiate the right side of the equation.

Tap for more steps...

Step 4.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x^{4}$ and $g (x) = {(1 - 2 x)}^{\frac{1}{2}}$ .

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

Step 4.2

Multiply the exponents in ${({(1 - 2 x)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 4.2.1

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

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

Step 4.2.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.2.2.1

Cancel the common factor.

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

Step 4.2.2.2

Rewrite the expression.

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

Step 4.3

Simplify.

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

Step 4.4

Differentiate using the Power Rule.

Tap for more steps...

Step 4.4.1

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

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

Step 4.4.2

Move $4$ to the left of ${(1 - 2 x)}^{\frac{1}{2}}$ .

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

Step 4.5

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

Tap for more steps...

Step 4.5.1

To apply the Chain Rule, set $u$ as $1 - 2 x$ .

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

Step 4.5.2

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

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

Step 4.5.3

Replace all occurrences of $u$ with $1 - 2 x$ .

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

Step 4.6

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

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

Step 4.7

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

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

Step 4.8

Combine the numerators over the common denominator.

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

Step 4.9

Simplify the numerator.

Tap for more steps...

Step 4.9.1

Multiply $- 1$ by $2$ .

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

Step 4.9.2

Subtract $2$ from $1$ .

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

Step 4.10

Combine fractions.

Tap for more steps...

Step 4.10.1

Move the negative in front of the fraction.

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

Step 4.10.2

Combine $\frac{1}{2}$ and ${(1 - 2 x)}^{- \frac{1}{2}}$ .

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

Step 4.10.3

Move ${(1 - 2 x)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 4.10.4

Combine $\frac{1}{2 {(1 - 2 x)}^{\frac{1}{2}}}$ and $x^{4}$ .

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

Step 4.11

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

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

Step 4.12

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{4 {(1 - 2 x)}^{\frac{1}{2}} x^{3} - \frac{x^{4}}{2 {(1 - 2 x)}^{\frac{1}{2}}} (0 + \frac{d}{d x} [- 2 x])}{1 - 2 x}$

Step 4.13

Add $0$ and $\frac{d}{d x} [- 2 x]$ .

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

Step 4.14

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

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

Step 4.15

Simplify terms.

Tap for more steps...

Step 4.15.1

Multiply $- 2$ by $- 1$ .

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

Step 4.15.2

Combine $2$ and $\frac{x^{4}}{2 {(1 - 2 x)}^{\frac{1}{2}}}$ .

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

Step 4.15.3

Cancel the common factor.

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

Step 4.15.4

Rewrite the expression.

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

Step 4.16

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

$\frac{4 {(1 - 2 x)}^{\frac{1}{2}} x^{3} + \frac{x^{4}}{{(1 - 2 x)}^{\frac{1}{2}}} \cdot 1}{1 - 2 x}$

Step 4.17

Simplify the expression.

Tap for more steps...

Step 4.17.1

Multiply $\frac{x^{4}}{{(1 - 2 x)}^{\frac{1}{2}}}$ by $1$ .

$\frac{4 {(1 - 2 x)}^{\frac{1}{2}} x^{3} + \frac{x^{4}}{{(1 - 2 x)}^{\frac{1}{2}}}}{1 - 2 x}$

Step 4.17.2

Reorder $1$ and $- 2 x$ .

$\frac{4 x^{3} {(- 2 x + 1)}^{\frac{1}{2}} + \frac{x^{4}}{{(- 2 x + 1)}^{\frac{1}{2}}}}{1 - 2 x}$

Step 4.18

To write $4 x^{3} {(- 2 x + 1)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(- 2 x + 1)}^{\frac{1}{2}}}{{(- 2 x + 1)}^{\frac{1}{2}}}$ .

$\frac{\frac{4 x^{3} {(- 2 x + 1)}^{\frac{1}{2}} {(- 2 x + 1)}^{\frac{1}{2}}}{{(- 2 x + 1)}^{\frac{1}{2}}} + \frac{x^{4}}{{(- 2 x + 1)}^{\frac{1}{2}}}}{1 - 2 x}$

Step 4.19

Combine the numerators over the common denominator.

$\frac{\frac{4 x^{3} {(- 2 x + 1)}^{\frac{1}{2}} {(- 2 x + 1)}^{\frac{1}{2}} + x^{4}}{{(- 2 x + 1)}^{\frac{1}{2}}}}{1 - 2 x}$

Step 4.20

Multiply ${(- 2 x + 1)}^{\frac{1}{2}}$ by ${(- 2 x + 1)}^{\frac{1}{2}}$ by adding the exponents.

Tap for more steps...

Step 4.20.1

Move ${(- 2 x + 1)}^{\frac{1}{2}}$ .

$\frac{\frac{4 x^{3} ({(- 2 x + 1)}^{\frac{1}{2}} {(- 2 x + 1)}^{\frac{1}{2}}) + x^{4}}{{(- 2 x + 1)}^{\frac{1}{2}}}}{1 - 2 x}$

Step 4.20.2

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

$\frac{\frac{4 x^{3} {(- 2 x + 1)}^{\frac{1}{2} + \frac{1}{2}} + x^{4}}{{(- 2 x + 1)}^{\frac{1}{2}}}}{1 - 2 x}$

Step 4.20.3

Combine the numerators over the common denominator.

$\frac{\frac{4 x^{3} {(- 2 x + 1)}^{\frac{1 + 1}{2}} + x^{4}}{{(- 2 x + 1)}^{\frac{1}{2}}}}{1 - 2 x}$

Step 4.20.4

Add $1$ and $1$ .

$\frac{\frac{4 x^{3} {(- 2 x + 1)}^{\frac{2}{2}} + x^{4}}{{(- 2 x + 1)}^{\frac{1}{2}}}}{1 - 2 x}$

Step 4.20.5

Divide $2$ by $2$ .

$\frac{\frac{4 x^{3} {(- 2 x + 1)}^{1} + x^{4}}{{(- 2 x + 1)}^{\frac{1}{2}}}}{1 - 2 x}$

Step 4.21

Simplify $4 x^{3} {(- 2 x + 1)}^{1}$ .

$\frac{\frac{4 x^{3} (- 2 x + 1) + x^{4}}{{(- 2 x + 1)}^{\frac{1}{2}}}}{1 - 2 x}$

Step 4.22

Rewrite $\frac{\frac{4 x^{3} (- 2 x + 1) + x^{4}}{{(- 2 x + 1)}^{\frac{1}{2}}}}{1 - 2 x}$ as a product.

$\frac{4 x^{3} (- 2 x + 1) + x^{4}}{{(- 2 x + 1)}^{\frac{1}{2}}} \cdot \frac{1}{1 - 2 x}$

Step 4.23

Multiply $\frac{4 x^{3} (- 2 x + 1) + x^{4}}{{(- 2 x + 1)}^{\frac{1}{2}}}$ by $\frac{1}{1 - 2 x}$ .

$\frac{4 x^{3} (- 2 x + 1) + x^{4}}{{(- 2 x + 1)}^{\frac{1}{2}} (1 - 2 x)}$

Step 4.24

Reorder terms.

$\frac{4 x^{3} (- 2 x + 1) + x^{4}}{{(- 2 x + 1)}^{\frac{1}{2}} (- 2 x + 1)}$

Step 4.25

Multiply ${(- 2 x + 1)}^{\frac{1}{2}}$ by $- 2 x + 1$ by adding the exponents.

Tap for more steps...

Step 4.25.1

Multiply ${(- 2 x + 1)}^{\frac{1}{2}}$ by $- 2 x + 1$ .

Tap for more steps...

Step 4.25.1.1

Raise $- 2 x + 1$ to the power of $1$ .

$\frac{4 x^{3} (- 2 x + 1) + x^{4}}{{(- 2 x + 1)}^{\frac{1}{2}} {(- 2 x + 1)}^{1}}$

Step 4.25.1.2

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

$\frac{4 x^{3} (- 2 x + 1) + x^{4}}{{(- 2 x + 1)}^{\frac{1}{2} + 1}}$

Step 4.25.2

Write $1$ as a fraction with a common denominator.

$\frac{4 x^{3} (- 2 x + 1) + x^{4}}{{(- 2 x + 1)}^{\frac{1}{2} + \frac{2}{2}}}$

Step 4.25.3

Combine the numerators over the common denominator.

$\frac{4 x^{3} (- 2 x + 1) + x^{4}}{{(- 2 x + 1)}^{\frac{1 + 2}{2}}}$

Step 4.25.4

Add $1$ and $2$ .

$\frac{4 x^{3} (- 2 x + 1) + x^{4}}{{(- 2 x + 1)}^{\frac{3}{2}}}$

Step 4.26

Simplify.

Tap for more steps...

Step 4.26.1

Apply the distributive property.

$\frac{4 x^{3} (- 2 x) + 4 x^{3} \cdot 1 + x^{4}}{{(- 2 x + 1)}^{\frac{3}{2}}}$

Step 4.26.2

Simplify the numerator.

Tap for more steps...

Step 4.26.2.1

Simplify each term.

Tap for more steps...

Step 4.26.2.1.1

Rewrite using the commutative property of multiplication.

$\frac{4 \cdot - 2 x^{3} x + 4 x^{3} \cdot 1 + x^{4}}{{(- 2 x + 1)}^{\frac{3}{2}}}$

Step 4.26.2.1.2

Multiply $x^{3}$ by $x$ by adding the exponents.

Tap for more steps...

Step 4.26.2.1.2.1

Move $x$ .

$\frac{4 \cdot - 2 (x \cdot x^{3}) + 4 x^{3} \cdot 1 + x^{4}}{{(- 2 x + 1)}^{\frac{3}{2}}}$

Step 4.26.2.1.2.2

Multiply $x$ by $x^{3}$ .

Tap for more steps...

Step 4.26.2.1.2.2.1

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

$\frac{4 \cdot - 2 (x^{1} x^{3}) + 4 x^{3} \cdot 1 + x^{4}}{{(- 2 x + 1)}^{\frac{3}{2}}}$

Step 4.26.2.1.2.2.2

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

$\frac{4 \cdot - 2 x^{1 + 3} + 4 x^{3} \cdot 1 + x^{4}}{{(- 2 x + 1)}^{\frac{3}{2}}}$

Step 4.26.2.1.2.3

Add $1$ and $3$ .

$\frac{4 \cdot - 2 x^{4} + 4 x^{3} \cdot 1 + x^{4}}{{(- 2 x + 1)}^{\frac{3}{2}}}$

Step 4.26.2.1.3

Multiply $4$ by $- 2$ .

$\frac{- 8 x^{4} + 4 x^{3} \cdot 1 + x^{4}}{{(- 2 x + 1)}^{\frac{3}{2}}}$

Step 4.26.2.1.4

Multiply $4$ by $1$ .

$\frac{- 8 x^{4} + 4 x^{3} + x^{4}}{{(- 2 x + 1)}^{\frac{3}{2}}}$

Step 4.26.2.2

Add $- 8 x^{4}$ and $x^{4}$ .

$\frac{- 7 x^{4} + 4 x^{3}}{{(- 2 x + 1)}^{\frac{3}{2}}}$

Step 4.26.3

Factor $x^{3}$ out of $- 7 x^{4} + 4 x^{3}$ .

Tap for more steps...

Step 4.26.3.1

Factor $x^{3}$ out of $- 7 x^{4}$ .

$\frac{x^{3} (- 7 x) + 4 x^{3}}{{(- 2 x + 1)}^{\frac{3}{2}}}$

Step 4.26.3.2

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

$\frac{x^{3} (- 7 x) + x^{3} \cdot 4}{{(- 2 x + 1)}^{\frac{3}{2}}}$

Step 4.26.3.3

Factor $x^{3}$ out of $x^{3} (- 7 x) + x^{3} \cdot 4$ .

$\frac{x^{3} (- 7 x + 4)}{{(- 2 x + 1)}^{\frac{3}{2}}}$

Step 4.26.4

Factor $- 1$ out of $- 7 x$ .

$\frac{x^{3} (- (7 x) + 4)}{{(- 2 x + 1)}^{\frac{3}{2}}}$

Step 4.26.5

Rewrite $4$ as $- 1 (- 4)$ .

$\frac{x^{3} (- (7 x) - 1 (- 4))}{{(- 2 x + 1)}^{\frac{3}{2}}}$

Step 4.26.6

Factor $- 1$ out of $- (7 x) - 1 (- 4)$ .

$\frac{x^{3} (- (7 x - 4))}{{(- 2 x + 1)}^{\frac{3}{2}}}$

Step 4.26.7

Rewrite $- (7 x - 4)$ as $- 1 (7 x - 4)$ .

$\frac{x^{3} (- 1 (7 x - 4))}{{(- 2 x + 1)}^{\frac{3}{2}}}$

Step 4.26.8

Move the negative in front of the fraction.

$- \frac{x^{3} (7 x - 4)}{{(- 2 x + 1)}^{\frac{3}{2}}}$

Step 5

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

$y' = - \frac{x^{3} (7 x - 4)}{{(- 2 x + 1)}^{\frac{3}{2}}}$

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = - \frac{x^{3} (7 x - 4)}{{(- 2 x + 1)}^{\frac{3}{2}}}$