Calculus Examples

$y = {(a + \frac{b}{x^{2}})}^{3}$

Step 1

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

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Step 1.1

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

$\frac{d}{d u} [u^{3}] \frac{d}{d x} [a + \frac{b}{x^{2}}]$

Step 1.2

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

$3 u^{2} \frac{d}{d x} [a + \frac{b}{x^{2}}]$

Step 1.3

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

$3 {(a + \frac{b}{x^{2}})}^{2} \frac{d}{d x} [a + \frac{b}{x^{2}}]$

Step 2

Differentiate.

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Step 2.1

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

$3 {(a + \frac{b}{x^{2}})}^{2} (\frac{d}{d x} [a] + \frac{d}{d x} [\frac{b}{x^{2}}])$

Step 2.2

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

$3 {(a + \frac{b}{x^{2}})}^{2} (0 + \frac{d}{d x} [\frac{b}{x^{2}}])$

Step 2.3

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

$3 {(a + \frac{b}{x^{2}})}^{2} \frac{d}{d x} [\frac{b}{x^{2}}]$

Step 2.4

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

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

Step 2.5

Apply basic rules of exponents.

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Step 2.5.1

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

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

Step 2.5.2

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

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Step 2.5.2.1

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

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

Step 2.5.2.2

Multiply $2$ by $- 1$ .

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

Step 2.6

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

$3 {(a + \frac{b}{x^{2}})}^{2} (b (- 2 x^{- 3}))$

Step 2.7

Multiply $- 2$ by $3$ .

$- 6 {(a + \frac{b}{x^{2}})}^{2} (b (x^{- 3}))$

Step 3

Simplify.

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Step 3.1

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

$- 6 {(a + \frac{b}{x^{2}})}^{2} (b \frac{1}{x^{3}})$

Step 3.2

Combine terms.

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Step 3.2.1

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

$- 6 {(a + \frac{b}{x^{2}})}^{2} \frac{b}{x^{3}}$

Step 3.2.2

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

$\frac{b \cdot - 6}{x^{3}} {(a + \frac{b}{x^{2}})}^{2}$

Step 3.2.3

Combine $\frac{b \cdot - 6}{x^{3}}$ and ${(a + \frac{b}{x^{2}})}^{2}$ .

$\frac{b \cdot - 6 {(a + \frac{b}{x^{2}})}^{2}}{x^{3}}$

Step 3.2.4

Move $- 6$ to the left of $b$ .

$\frac{- 6 \cdot b {(a + \frac{b}{x^{2}})}^{2}}{x^{3}}$

Step 3.2.5

Move the negative in front of the fraction.

$- \frac{6 b {(a + \frac{b}{x^{2}})}^{2}}{x^{3}}$

Step 3.3

Simplify the numerator.

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Step 3.3.1

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

$- \frac{6 b {(\frac{a x^{2}}{x^{2}} + \frac{b}{x^{2}})}^{2}}{x^{3}}$

Step 3.3.2

Combine the numerators over the common denominator.

$- \frac{6 b {(\frac{a x^{2} + b}{x^{2}})}^{2}}{x^{3}}$

Step 3.3.3

Apply the product rule to $\frac{a x^{2} + b}{x^{2}}$ .

$- \frac{6 b \frac{{(a x^{2} + b)}^{2}}{{(x^{2})}^{2}}}{x^{3}}$

Step 3.3.4

Combine exponents.

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Step 3.3.4.1

Combine $6$ and $\frac{{(a x^{2} + b)}^{2}}{{(x^{2})}^{2}}$ .

$- \frac{b \frac{6 {(a x^{2} + b)}^{2}}{{(x^{2})}^{2}}}{x^{3}}$

Step 3.3.4.2

Combine $b$ and $\frac{6 {(a x^{2} + b)}^{2}}{{(x^{2})}^{2}}$ .

$- \frac{\frac{b (6 {(a x^{2} + b)}^{2})}{{(x^{2})}^{2}}}{x^{3}}$

Step 3.3.5

Remove unnecessary parentheses.

$- \frac{\frac{b \cdot 6 {(a x^{2} + b)}^{2}}{{(x^{2})}^{2}}}{x^{3}}$

Step 3.3.6

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

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Step 3.3.6.1

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

$- \frac{\frac{b \cdot 6 {(a x^{2} + b)}^{2}}{x^{2 \cdot 2}}}{x^{3}}$

Step 3.3.6.2

Multiply $2$ by $2$ .

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

Step 3.3.7

Move $6$ to the left of $b$ .

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

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.5

Combine.

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

Step 3.6

Multiply $x^{4}$ by $x^{3}$ by adding the exponents.

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Step 3.6.1

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

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

Step 3.6.2

Add $4$ and $3$ .

$- \frac{6 b {(a x^{2} + b)}^{2} \cdot 1}{x^{7}}$

Step 3.7

Multiply $6 b {(a x^{2} + b)}^{2}$ by $1$ .

$- \frac{6 b {(a x^{2} + b)}^{2}}{x^{7}}$