Calculus Examples

$y = x^{2} \sqrt{4 x - 3}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 x - 3}$ as ${(4 x - 3)}^{\frac{1}{2}}$ .

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

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2}$ and $g (x) = {(4 x - 3)}^{\frac{1}{2}}$ .

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

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 4 x - 3$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $4 x - 3$ .

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

Step 3.2

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

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

Step 3.3

Replace all occurrences of $u$ with $4 x - 3$ .

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

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

Step 7.2

Subtract $2$ from $1$ .

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

Step 8

Combine fractions.

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Step 8.1

Move the negative in front of the fraction.

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

Step 8.2

Combine $\frac{1}{2}$ and ${(4 x - 3)}^{- \frac{1}{2}}$ .

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

Step 8.3

Move ${(4 x - 3)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 8.4

Combine $\frac{1}{2 {(4 x - 3)}^{\frac{1}{2}}}$ and $x^{2}$ .

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

Multiply $4$ by $1$ .

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

Step 13

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

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

Step 14

Simplify terms.

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Step 14.1

Add $4$ and $0$ .

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

Step 14.2

Combine $\frac{x^{2}}{2 {(4 x - 3)}^{\frac{1}{2}}}$ and $4$ .

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

Step 14.3

Move $4$ to the left of $x^{2}$ .

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

Step 14.4

Factor $2$ out of $4 x^{2}$ .

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

Step 15

Cancel the common factors.

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Step 15.1

Factor $2$ out of $2 {(4 x - 3)}^{\frac{1}{2}}$ .

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

Step 15.2

Cancel the common factor.

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

Step 15.3

Rewrite the expression.

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

Step 16

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

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

Step 17

Move $2$ to the left of ${(4 x - 3)}^{\frac{1}{2}}$ .

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

Step 18

Combine $\frac{2 x^{2}}{{(4 x - 3)}^{\frac{1}{2}}}$ and $2 {(4 x - 3)}^{\frac{1}{2}} x$ using a common denominator.

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Step 18.1

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

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

Step 18.2

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

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

Step 18.3

Combine the numerators over the common denominator.

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

Step 19

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

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Step 19.1

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

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

Step 19.2

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

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

Step 19.3

Combine the numerators over the common denominator.

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

Step 19.4

Add $1$ and $1$ .

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

Step 19.5

Divide $2$ by $2$ .

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

Step 20

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

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

Step 21

Simplify.

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Step 21.1

Apply the distributive property.

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

Step 21.2

Simplify the numerator.

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Step 21.2.1

Simplify each term.

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Step 21.2.1.1

Rewrite using the commutative property of multiplication.

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

Step 21.2.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 21.2.1.2.1

Move $x$ .

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

Step 21.2.1.2.2

Multiply $x$ by $x$ .

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

Step 21.2.1.3

Multiply $2$ by $4$ .

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

Step 21.2.1.4

Multiply $- 3$ by $2$ .

$\frac{2 x^{2} + 8 x^{2} - 6 x}{{(4 x - 3)}^{\frac{1}{2}}}$

Step 21.2.2

Add $2 x^{2}$ and $8 x^{2}$ .

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

Step 21.3

Factor $2 x$ out of $10 x^{2} - 6 x$ .

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Step 21.3.1

Factor $2 x$ out of $10 x^{2}$ .

$\frac{2 x (5 x) - 6 x}{{(4 x - 3)}^{\frac{1}{2}}}$

Step 21.3.2

Factor $2 x$ out of $- 6 x$ .

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

Step 21.3.3

Factor $2 x$ out of $2 x (5 x) + 2 x (- 3)$ .

$\frac{2 x (5 x - 3)}{{(4 x - 3)}^{\frac{1}{2}}}$