Calculus Examples

$5 x^{2} - 4 = - 4 y^{2}$

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $5 x^{2} - 4$ with respect to $x$ is $\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [- 4]$ .

$\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [- 4]$

Step 2.2

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

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Step 2.2.1

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

$5 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4]$

Step 2.2.2

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

$5 (2 x) + \frac{d}{d x} [- 4]$

Step 2.2.3

Multiply $2$ by $5$ .

$10 x + \frac{d}{d x} [- 4]$

Step 2.3

Differentiate using the Constant Rule.

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Step 2.3.1

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

$10 x + 0$

Step 2.3.2

Add $10 x$ and $0$ .

$10 x$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

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 3.2.1

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

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

Step 3.2.2

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

$- 4 (2 u \frac{d}{d x} [y])$

Step 3.2.3

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

$- 4 (2 y \frac{d}{d x} [y])$

Step 3.3

Multiply $2$ by $- 4$ .

$- 8 (y \frac{d}{d x} [y])$

Step 3.4

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

$- 8 y y'$

Step 4

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

$10 x = - 8 y y'$

Step 5

Solve for $y'$ .

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Step 5.1

Rewrite the equation as $- 8 y y' = 10 x$ .

$- 8 y y' = 10 x$

Step 5.2

Divide each term in $- 8 y y' = 10 x$ by $- 8 y$ and simplify.

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Step 5.2.1

Divide each term in $- 8 y y' = 10 x$ by $- 8 y$ .

$\frac{- 8 y y'}{- 8 y} = \frac{10 x}{- 8 y}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $- 8$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{- 8 y y'}{- 8 y} = \frac{10 x}{- 8 y}$

Step 5.2.2.1.2

Rewrite the expression.

$\frac{y y'}{y} = \frac{10 x}{- 8 y}$

Step 5.2.2.2

Cancel the common factor of $y$ .

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Step 5.2.2.2.1

Cancel the common factor.

$\frac{y y'}{y} = \frac{10 x}{- 8 y}$

Step 5.2.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{10 x}{- 8 y}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Cancel the common factor of $10$ and $- 8$ .

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Step 5.2.3.1.1

Factor $2$ out of $10 x$ .

$y' = \frac{2 (5 x)}{- 8 y}$

Step 5.2.3.1.2

Cancel the common factors.

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Step 5.2.3.1.2.1

Factor $2$ out of $- 8 y$ .

$y' = \frac{2 (5 x)}{2 (- 4 y)}$

Step 5.2.3.1.2.2

Cancel the common factor.

$y' = \frac{2 (5 x)}{2 (- 4 y)}$

Step 5.2.3.1.2.3

Rewrite the expression.

$y' = \frac{5 x}{- 4 y}$

Step 5.2.3.2

Move the negative in front of the fraction.

$y' = - \frac{5 x}{4 y}$

Step 6

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

$\frac{d y}{d x} = - \frac{5 x}{4 y}$