Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

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

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

Apply basic rules of exponents.

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Step 3.2.1

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

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

Step 3.2.2

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

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Step 3.2.2.1

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

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

Step 3.2.2.2

Multiply $2$ by $- 1$ .

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

Step 3.3

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

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

Step 3.4

Simplify terms.

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Step 3.4.1

Combine $- 2$ and $\frac{3}{4}$ .

$\frac{- 2 \cdot 3}{4} x^{- 3}$

Step 3.4.2

Multiply $- 2$ by $3$ .

$\frac{- 6}{4} x^{- 3}$

Step 3.4.3

Combine $\frac{- 6}{4}$ and $x^{- 3}$ .

$\frac{- 6 x^{- 3}}{4}$

Step 3.4.4

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

$\frac{- 6}{4 x^{3}}$

Step 3.4.5

Cancel the common factor of $- 6$ and $4$ .

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Step 3.4.5.1

Factor $2$ out of $- 6$ .

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

Step 3.4.5.2

Cancel the common factors.

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Step 3.4.5.2.1

Factor $2$ out of $4 x^{3}$ .

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

Step 3.4.5.2.2

Cancel the common factor.

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

Step 3.4.5.2.3

Rewrite the expression.

$\frac{- 3}{2 x^{3}}$

Step 3.4.6

Move the negative in front of the fraction.

$- \frac{3}{2 x^{3}}$

Step 4

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

$y' = - \frac{3}{2 x^{3}}$

Step 5

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

$\frac{d y}{d x} = - \frac{3}{2 x^{3}}$