Calculus Examples

$y = 4 {(\sqrt{x} + 1)}^{5}$

Step 1

Differentiate using the Constant Multiple Rule.

Tap for more steps...

Step 1.1

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

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

Step 1.2

Since $4$ is constant with respect to $x$ , the derivative of $4 {(x^{\frac{1}{2}} + 1)}^{5}$ with respect to $x$ is $4 \frac{d}{d x} [{(x^{\frac{1}{2}} + 1)}^{5}]$ .

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{5}$ and $g (x) = x^{\frac{1}{2}} + 1$ .

Tap for more steps...

Step 2.1

To apply the Chain Rule, set $u$ as $x^{\frac{1}{2}} + 1$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $x^{\frac{1}{2}} + 1$ .

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

Step 3

Differentiate.

Tap for more steps...

Step 3.1

Multiply $5$ by $4$ .

$20 ({(x^{\frac{1}{2}} + 1)}^{4} \frac{d}{d x} [x^{\frac{1}{2}} + 1])$

Step 3.2

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

$20 {(x^{\frac{1}{2}} + 1)}^{4} (\frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [1])$

Step 3.3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

Tap for more steps...

Step 7.1

Multiply $- 1$ by $2$ .

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

Step 7.2

Subtract $2$ from $1$ .

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

Step 8

Combine fractions.

Tap for more steps...

Step 8.1

Move the negative in front of the fraction.

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

Step 8.2

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

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

Step 8.3

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

$20 {(x^{\frac{1}{2}} + 1)}^{4} (\frac{1}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [1])$

Step 9

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

$20 {(x^{\frac{1}{2}} + 1)}^{4} (\frac{1}{2 x^{\frac{1}{2}}} + 0)$

Step 10

Simplify terms.

Tap for more steps...

Step 10.1

Add $\frac{1}{2 x^{\frac{1}{2}}}$ and $0$ .

$20 {(x^{\frac{1}{2}} + 1)}^{4} \frac{1}{2 x^{\frac{1}{2}}}$

Step 10.2

Combine $\frac{1}{2 x^{\frac{1}{2}}}$ and $20$ .

$\frac{20}{2 x^{\frac{1}{2}}} {(x^{\frac{1}{2}} + 1)}^{4}$

Step 10.3

Combine $\frac{20}{2 x^{\frac{1}{2}}}$ and ${(x^{\frac{1}{2}} + 1)}^{4}$ .

$\frac{20 {(x^{\frac{1}{2}} + 1)}^{4}}{2 x^{\frac{1}{2}}}$

Step 10.4

Factor $2$ out of $20 {(x^{\frac{1}{2}} + 1)}^{4}$ .

$\frac{2 (10 {(x^{\frac{1}{2}} + 1)}^{4})}{2 x^{\frac{1}{2}}}$

Step 11

Cancel the common factors.

Tap for more steps...

Step 11.1

Factor $2$ out of $2 x^{\frac{1}{2}}$ .

$\frac{2 (10 {(x^{\frac{1}{2}} + 1)}^{4})}{2 (x^{\frac{1}{2}})}$

Step 11.2

Cancel the common factor.

$\frac{2 (10 {(x^{\frac{1}{2}} + 1)}^{4})}{2 x^{\frac{1}{2}}}$

Step 11.3

Rewrite the expression.

$\frac{10 {(x^{\frac{1}{2}} + 1)}^{4}}{x^{\frac{1}{2}}}$

Step 12

Simplify.

Tap for more steps...

Step 12.1

Simplify the numerator.

Tap for more steps...

Step 12.1.1

Use the Binomial Theorem.

$\frac{10 ({(x^{\frac{1}{2}})}^{4} + 4 {(x^{\frac{1}{2}})}^{3} \cdot 1 + 6 {(x^{\frac{1}{2}})}^{2} \cdot 1^{2} + 4 x^{\frac{1}{2}} \cdot 1^{3} + 1^{4})}{x^{\frac{1}{2}}}$

Step 12.1.2

Simplify each term.

Tap for more steps...

Step 12.1.2.1

Multiply the exponents in ${(x^{\frac{1}{2}})}^{4}$ .

Tap for more steps...

Step 12.1.2.1.1

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

$\frac{10 (x^{\frac{1}{2} \cdot 4} + 4 {(x^{\frac{1}{2}})}^{3} \cdot 1 + 6 {(x^{\frac{1}{2}})}^{2} \cdot 1^{2} + 4 x^{\frac{1}{2}} \cdot 1^{3} + 1^{4})}{x^{\frac{1}{2}}}$

Step 12.1.2.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 12.1.2.1.2.1

Factor $2$ out of $4$ .

$\frac{10 (x^{\frac{1}{2} \cdot (2 (2))} + 4 {(x^{\frac{1}{2}})}^{3} \cdot 1 + 6 {(x^{\frac{1}{2}})}^{2} \cdot 1^{2} + 4 x^{\frac{1}{2}} \cdot 1^{3} + 1^{4})}{x^{\frac{1}{2}}}$

Step 12.1.2.1.2.2

Cancel the common factor.

$\frac{10 (x^{\frac{1}{2} \cdot (2 \cdot 2)} + 4 {(x^{\frac{1}{2}})}^{3} \cdot 1 + 6 {(x^{\frac{1}{2}})}^{2} \cdot 1^{2} + 4 x^{\frac{1}{2}} \cdot 1^{3} + 1^{4})}{x^{\frac{1}{2}}}$

Step 12.1.2.1.2.3

Rewrite the expression.

$\frac{10 (x^{2} + 4 {(x^{\frac{1}{2}})}^{3} \cdot 1 + 6 {(x^{\frac{1}{2}})}^{2} \cdot 1^{2} + 4 x^{\frac{1}{2}} \cdot 1^{3} + 1^{4})}{x^{\frac{1}{2}}}$

Step 12.1.2.2

Multiply the exponents in ${(x^{\frac{1}{2}})}^{3}$ .

Tap for more steps...

Step 12.1.2.2.1

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

$\frac{10 (x^{2} + 4 x^{\frac{1}{2} \cdot 3} \cdot 1 + 6 {(x^{\frac{1}{2}})}^{2} \cdot 1^{2} + 4 x^{\frac{1}{2}} \cdot 1^{3} + 1^{4})}{x^{\frac{1}{2}}}$

Step 12.1.2.2.2

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

$\frac{10 (x^{2} + 4 x^{\frac{3}{2}} \cdot 1 + 6 {(x^{\frac{1}{2}})}^{2} \cdot 1^{2} + 4 x^{\frac{1}{2}} \cdot 1^{3} + 1^{4})}{x^{\frac{1}{2}}}$

Step 12.1.2.3

Multiply $4$ by $1$ .

$\frac{10 (x^{2} + 4 x^{\frac{3}{2}} + 6 {(x^{\frac{1}{2}})}^{2} \cdot 1^{2} + 4 x^{\frac{1}{2}} \cdot 1^{3} + 1^{4})}{x^{\frac{1}{2}}}$

Step 12.1.2.4

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

Tap for more steps...

Step 12.1.2.4.1

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

$\frac{10 (x^{2} + 4 x^{\frac{3}{2}} + 6 x^{\frac{1}{2} \cdot 2} \cdot 1^{2} + 4 x^{\frac{1}{2}} \cdot 1^{3} + 1^{4})}{x^{\frac{1}{2}}}$

Step 12.1.2.4.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 12.1.2.4.2.1

Cancel the common factor.

$\frac{10 (x^{2} + 4 x^{\frac{3}{2}} + 6 x^{\frac{1}{2} \cdot 2} \cdot 1^{2} + 4 x^{\frac{1}{2}} \cdot 1^{3} + 1^{4})}{x^{\frac{1}{2}}}$

Step 12.1.2.4.2.2

Rewrite the expression.

$\frac{10 (x^{2} + 4 x^{\frac{3}{2}} + 6 x^{1} \cdot 1^{2} + 4 x^{\frac{1}{2}} \cdot 1^{3} + 1^{4})}{x^{\frac{1}{2}}}$

Step 12.1.2.5

Simplify.

$\frac{10 (x^{2} + 4 x^{\frac{3}{2}} + 6 x \cdot 1^{2} + 4 x^{\frac{1}{2}} \cdot 1^{3} + 1^{4})}{x^{\frac{1}{2}}}$

Step 12.1.2.6

One to any power is one.

$\frac{10 (x^{2} + 4 x^{\frac{3}{2}} + 6 x \cdot 1 + 4 x^{\frac{1}{2}} \cdot 1^{3} + 1^{4})}{x^{\frac{1}{2}}}$

Step 12.1.2.7

Multiply $6$ by $1$ .

$\frac{10 (x^{2} + 4 x^{\frac{3}{2}} + 6 x + 4 x^{\frac{1}{2}} \cdot 1^{3} + 1^{4})}{x^{\frac{1}{2}}}$

Step 12.1.2.8

One to any power is one.

$\frac{10 (x^{2} + 4 x^{\frac{3}{2}} + 6 x + 4 x^{\frac{1}{2}} \cdot 1 + 1^{4})}{x^{\frac{1}{2}}}$

Step 12.1.2.9

Multiply $4$ by $1$ .

$\frac{10 (x^{2} + 4 x^{\frac{3}{2}} + 6 x + 4 x^{\frac{1}{2}} + 1^{4})}{x^{\frac{1}{2}}}$

Step 12.1.2.10

One to any power is one.

$\frac{10 (x^{2} + 4 x^{\frac{3}{2}} + 6 x + 4 x^{\frac{1}{2}} + 1)}{x^{\frac{1}{2}}}$

Step 12.1.3

Apply the distributive property.

$\frac{10 x^{2} + 10 (4 x^{\frac{3}{2}}) + 10 (6 x) + 10 (4 x^{\frac{1}{2}}) + 10 \cdot 1}{x^{\frac{1}{2}}}$

Step 12.1.4

Simplify.

Tap for more steps...

Step 12.1.4.1

Multiply $4$ by $10$ .

$\frac{10 x^{2} + 40 x^{\frac{3}{2}} + 10 (6 x) + 10 (4 x^{\frac{1}{2}}) + 10 \cdot 1}{x^{\frac{1}{2}}}$

Step 12.1.4.2

Multiply $6$ by $10$ .

$\frac{10 x^{2} + 40 x^{\frac{3}{2}} + 60 x + 10 (4 x^{\frac{1}{2}}) + 10 \cdot 1}{x^{\frac{1}{2}}}$

Step 12.1.4.3

Multiply $4$ by $10$ .

$\frac{10 x^{2} + 40 x^{\frac{3}{2}} + 60 x + 40 x^{\frac{1}{2}} + 10 \cdot 1}{x^{\frac{1}{2}}}$

Step 12.1.4.4

Multiply $10$ by $1$ .

$\frac{10 x^{2} + 40 x^{\frac{3}{2}} + 60 x + 40 x^{\frac{1}{2}} + 10}{x^{\frac{1}{2}}}$

Step 12.2

Reorder terms.

$\frac{10 x^{2} + 60 x + 40 x^{\frac{3}{2}} + 40 x^{\frac{1}{2}} + 10}{x^{\frac{1}{2}}}$