Calculus Examples

$y = \frac{3}{4 + x^{2}}$

Differentiate both sides of the equation.

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

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

$y'$

Differentiate the right side of the equation.

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Differentiate using the Constant Multiple Rule.

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Since $3$ is constant with respect to $x$ , the derivative of $\frac{3}{4 + x^{2}}$ with respect to $x$ is $3 \frac{d}{d x} [\frac{1}{4 + x^{2}}]$ .

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

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

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

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

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To apply the Chain Rule, set $u$ as $4 + x^{2}$ .

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

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

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

Replace all occurrences of $u$ with $4 + x^{2}$ .

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

Differentiate.

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Multiply $- 1$ by $3$ .

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

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

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

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

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

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

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

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

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

Multiply $2$ by $- 3$ .

$- 6 {(4 + x^{2})}^{- 2} x$

Simplify.

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

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

Combine terms.

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Combine $- 6$ and $\frac{1}{{(4 + x^{2})}^{2}}$ .

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

Move the negative in front of the fraction.

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

Combine $x$ and $\frac{6}{{(4 + x^{2})}^{2}}$ .

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

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

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

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

$y' = - \frac{6 x}{{(4 + x^{2})}^{2}}$

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

$\frac{d y}{d x} = - \frac{6 x}{{(4 + x^{2})}^{2}}$