Calculus Examples

$\frac{7 x^{2}}{{(7 x + 13 y)}^{2}}$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Since $7 x^{2}$ is constant with respect to $y$ , the derivative of $\frac{7 x^{2}}{{(7 x + 13 y)}^{2}}$ with respect to $y$ is $7 x^{2} \frac{d}{d y} [\frac{1}{{(7 x + 13 y)}^{2}}]$ .

$7 x^{2} \frac{d}{d y} [\frac{1}{{(7 x + 13 y)}^{2}}]$

Step 1.2

Apply basic rules of exponents.

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Step 1.2.1

Rewrite $\frac{1}{{(7 x + 13 y)}^{2}}$ as ${({(7 x + 13 y)}^{2})}^{- 1}$ .

$7 x^{2} \frac{d}{d y} [{({(7 x + 13 y)}^{2})}^{- 1}]$

Step 1.2.2

Multiply the exponents in ${({(7 x + 13 y)}^{2})}^{- 1}$ .

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Step 1.2.2.1

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

$7 x^{2} \frac{d}{d y} [{(7 x + 13 y)}^{2 \cdot - 1}]$

Step 1.2.2.2

Multiply $2$ by $- 1$ .

$7 x^{2} \frac{d}{d y} [{(7 x + 13 y)}^{- 2}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{- 2}$ and $g (y) = 7 x + 13 y$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $7 x + 13 y$ .

$7 x^{2} (\frac{d}{d u} [u^{- 2}] \frac{d}{d y} [7 x + 13 y])$

Step 2.2

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

$7 x^{2} (- 2 u^{- 3} \frac{d}{d y} [7 x + 13 y])$

Step 2.3

Replace all occurrences of $u$ with $7 x + 13 y$ .

$7 x^{2} (- 2 {(7 x + 13 y)}^{- 3} \frac{d}{d y} [7 x + 13 y])$

Step 3

Differentiate.

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Step 3.1

Multiply $- 2$ by $7$ .

$- 14 x^{2} ({(7 x + 13 y)}^{- 3} \frac{d}{d y} [7 x + 13 y])$

Step 3.2

By the Sum Rule, the derivative of $7 x + 13 y$ with respect to $y$ is $\frac{d}{d y} [7 x] + \frac{d}{d y} [13 y]$ .

$- 14 x^{2} ({(7 x + 13 y)}^{- 3} (\frac{d}{d y} [7 x] + \frac{d}{d y} [13 y]))$

Step 3.3

Since $7 x$ is constant with respect to $y$ , the derivative of $7 x$ with respect to $y$ is $0$ .

$- 14 x^{2} ({(7 x + 13 y)}^{- 3} (0 + \frac{d}{d y} [13 y]))$

Step 3.4

Add $0$ and $\frac{d}{d y} [13 y]$ .

$- 14 x^{2} ({(7 x + 13 y)}^{- 3} \frac{d}{d y} [13 y])$

Step 3.5

Since $13$ is constant with respect to $y$ , the derivative of $13 y$ with respect to $y$ is $13 \frac{d}{d y} [y]$ .

$- 14 x^{2} ({(7 x + 13 y)}^{- 3} (13 \frac{d}{d y} [y]))$

Step 3.6

Simplify the expression.

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Step 3.6.1

Move $13$ to the left of ${(7 x + 13 y)}^{- 3}$ .

$- 14 x^{2} (13 \cdot {(7 x + 13 y)}^{- 3} \frac{d}{d y} [y])$

Step 3.6.2

Multiply $13$ by $- 14$ .

$- 182 x^{2} ({(7 x + 13 y)}^{- 3} \frac{d}{d y} [y])$

Step 3.7

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

$- 182 x^{2} ({(7 x + 13 y)}^{- 3} \cdot 1)$

Step 3.8

Multiply ${(7 x + 13 y)}^{- 3}$ by $1$ .

$- 182 x^{2} {(7 x + 13 y)}^{- 3}$

Step 4

Simplify.

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Step 4.1

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

$- 182 x^{2} \frac{1}{{(7 x + 13 y)}^{3}}$

Step 4.2

Combine terms.

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Step 4.2.1

Combine $- 182$ and $\frac{1}{{(7 x + 13 y)}^{3}}$ .

$x^{2} \frac{- 182}{{(7 x + 13 y)}^{3}}$

Step 4.2.2

Combine $x^{2}$ and $\frac{- 182}{{(7 x + 13 y)}^{3}}$ .

$\frac{x^{2} \cdot - 182}{{(7 x + 13 y)}^{3}}$

Step 4.2.3

Move $- 182$ to the left of $x^{2}$ .

$\frac{- 182 \cdot x^{2}}{{(7 x + 13 y)}^{3}}$

Step 4.2.4

Move the negative in front of the fraction.

$- \frac{182 x^{2}}{{(7 x + 13 y)}^{3}}$