Calculus Examples

$y^{2} - x^{3} = 22$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y^{2} - x^{3}) = \frac{d}{d x} (22)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $y^{2} - x^{3}$ with respect to $x$ is $\frac{d}{d x} [y^{2}] + \frac{d}{d x} [- x^{3}]$ .

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

Step 2.2

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

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Step 2.2.1

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

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Step 2.2.1.1

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

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

Step 2.2.1.2

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

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

Step 2.2.1.3

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

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

Step 2.2.2

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

$2 y y' + \frac{d}{d x} [- x^{3}]$

Step 2.3

Evaluate $\frac{d}{d x} [- x^{3}]$ .

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Step 2.3.1

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

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

Step 2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$2 y y' - (3 x^{2})$

Step 2.3.3

Multiply $3$ by $- 1$ .

$2 y y' - 3 x^{2}$

Step 2.4

Reorder terms.

$- 3 x^{2} + 2 y y'$

Step 3

Since $22$ is constant with respect to $x$ , the derivative of $22$ with respect to $x$ is $0$ .

$0$

Step 4

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

$- 3 x^{2} + 2 y y' = 0$

Step 5

Solve for $y'$ .

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Step 5.1

Add $3 x^{2}$ to both sides of the equation.

$2 y y' = 3 x^{2}$

Step 5.2

Divide each term in $2 y y' = 3 x^{2}$ by $2 y$ and simplify.

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Step 5.2.1

Divide each term in $2 y y' = 3 x^{2}$ by $2 y$ .

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $2$ .

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Step 5.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.1.2

Rewrite the expression.

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

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.

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

Step 5.2.2.2.2

Divide $y'$ by $1$ .

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

Step 6

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

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