Calculus Examples

$5 y^{2} = \frac{4 x - 3}{4 x + 3}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (5 y^{2}) = \frac{d}{d x} (\frac{4 x - 3}{4 x + 3})$

Step 2

Differentiate the left side of the equation.

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Step 2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 y^{2}$ with respect to $x$ is $5 \frac{d}{d x} [y^{2}]$ .

$5 \frac{d}{d x} [y^{2}]$

Step 2.2

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

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Step 2.2.1

To apply the Chain Rule, set $u$ as $y$ .

$5 (\frac{d}{d u} [u^{2}] \frac{d}{d x} [y])$

Step 2.2.2

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

$5 (2 u \frac{d}{d x} [y])$

Step 2.2.3

Replace all occurrences of $u$ with $y$ .

$5 (2 y \frac{d}{d x} [y])$

Step 2.3

Multiply $2$ by $5$ .

$10 (y \frac{d}{d x} [y])$

Step 2.4

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$10 y y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

Differentiate.

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

Multiply $4$ by $1$ .

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

Step 3.2.5

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

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

Step 3.2.6

Simplify the expression.

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Step 3.2.6.1

Add $4$ and $0$ .

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

Step 3.2.6.2

Move $4$ to the left of $4 x + 3$ .

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

Step 3.2.7

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

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

Step 3.2.8

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

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

Step 3.2.9

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

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

Step 3.2.10

Multiply $4$ by $1$ .

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

Step 3.2.11

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

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

Step 3.2.12

Simplify the expression.

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Step 3.2.12.1

Add $4$ and $0$ .

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

Step 3.2.12.2

Multiply $4$ by $- 1$ .

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

Step 3.3

Simplify.

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Step 3.3.1

Apply the distributive property.

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

Step 3.3.2

Apply the distributive property.

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

Step 3.3.3

Simplify the numerator.

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Step 3.3.3.1

Combine the opposite terms in $4 (4 x) + 4 \cdot 3 - 4 (4 x) - 4 \cdot - 3$ .

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Step 3.3.3.1.1

Subtract $4 (4 x)$ from $4 (4 x)$ .

$\frac{4 \cdot 3 + 0 - 4 \cdot - 3}{{(4 x + 3)}^{2}}$

Step 3.3.3.1.2

Add $4 \cdot 3$ and $0$ .

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

Step 3.3.3.2

Simplify each term.

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Step 3.3.3.2.1

Multiply $4$ by $3$ .

$\frac{12 - 4 \cdot - 3}{{(4 x + 3)}^{2}}$

Step 3.3.3.2.2

Multiply $- 4$ by $- 3$ .

$\frac{12 + 12}{{(4 x + 3)}^{2}}$

Step 3.3.3.3

Add $12$ and $12$ .

$\frac{24}{{(4 x + 3)}^{2}}$

Step 4

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

$10 y y' = \frac{24}{{(4 x + 3)}^{2}}$

Step 5

Divide each term in $10 y y' = \frac{24}{{(4 x + 3)}^{2}}$ by $10 y$ and simplify.

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Step 5.1

Divide each term in $10 y y' = \frac{24}{{(4 x + 3)}^{2}}$ by $10 y$ .

$\frac{10 y y'}{10 y} = \frac{\frac{24}{{(4 x + 3)}^{2}}}{10 y}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $10$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{10 y y'}{10 y} = \frac{\frac{24}{{(4 x + 3)}^{2}}}{10 y}$

Step 5.2.1.2

Rewrite the expression.

$\frac{y y'}{y} = \frac{\frac{24}{{(4 x + 3)}^{2}}}{10 y}$

Step 5.2.2

Cancel the common factor of $y$ .

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Step 5.2.2.1

Cancel the common factor.

$\frac{y y'}{y} = \frac{\frac{24}{{(4 x + 3)}^{2}}}{10 y}$

Step 5.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{\frac{24}{{(4 x + 3)}^{2}}}{10 y}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Multiply the numerator by the reciprocal of the denominator.

$y' = \frac{24}{{(4 x + 3)}^{2}} \cdot \frac{1}{10 y}$

Step 5.3.2

Combine.

$y' = \frac{24 \cdot 1}{{(4 x + 3)}^{2} (10 y)}$

Step 5.3.3

Cancel the common factor of $24$ and $10$ .

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Step 5.3.3.1

Factor $2$ out of $24 \cdot 1$ .

$y' = \frac{2 (12 \cdot 1)}{{(4 x + 3)}^{2} (10 y)}$

Step 5.3.3.2

Cancel the common factors.

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Step 5.3.3.2.1

Factor $2$ out of ${(4 x + 3)}^{2} (10 y)$ .

$y' = \frac{2 (12 \cdot 1)}{2 ({(4 x + 3)}^{2} (5 y))}$

Step 5.3.3.2.2

Cancel the common factor.

$y' = \frac{2 (12 \cdot 1)}{2 ({(4 x + 3)}^{2} (5 y))}$

Step 5.3.3.2.3

Rewrite the expression.

$y' = \frac{12 \cdot 1}{{(4 x + 3)}^{2} (5 y)}$

Step 5.3.4

Simplify the expression.

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Step 5.3.4.1

Multiply $12$ by $1$ .

$y' = \frac{12}{{(4 x + 3)}^{2} \cdot 5 y}$

Step 5.3.4.2

Move $5$ to the left of ${(4 x + 3)}^{2}$ .

$y' = \frac{12}{5 \cdot {(4 x + 3)}^{2} y}$

Step 5.3.4.3

Reorder factors in $\frac{12}{5 {(4 x + 3)}^{2} y}$ .

$y' = \frac{12}{5 y {(4 x + 3)}^{2}}$

Step 6

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

$\frac{d y}{d x} = \frac{12}{5 y {(4 x + 3)}^{2}}$