Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

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

$2 x$

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $4 y^{3} + 5 y^{2}$ with respect to $x$ is $\frac{d}{d x} [4 y^{3}] + \frac{d}{d x} [5 y^{2}]$ .

$\frac{d}{d x} [4 y^{3}] + \frac{d}{d x} [5 y^{2}]$

Step 3.2

Evaluate $\frac{d}{d x} [4 y^{3}]$ .

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Step 3.2.1

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

$4 \frac{d}{d x} [y^{3}] + \frac{d}{d x} [5 y^{2}]$

Step 3.2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = y$ .

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Step 3.2.2.1

To apply the Chain Rule, set $u_{1}$ as $y$ .

$4 (\frac{d}{d u_{1}} [{u_{1}}^{3}] \frac{d}{d x} [y]) + \frac{d}{d x} [5 y^{2}]$

Step 3.2.2.2

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

$4 (3 {u_{1}}^{2} \frac{d}{d x} [y]) + \frac{d}{d x} [5 y^{2}]$

Step 3.2.2.3

Replace all occurrences of $u_{1}$ with $y$ .

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

Step 3.2.3

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

$4 (3 y^{2} y') + \frac{d}{d x} [5 y^{2}]$

Step 3.2.4

Multiply $3$ by $4$ .

$12 y^{2} y' + \frac{d}{d x} [5 y^{2}]$

Step 3.3

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

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Step 3.3.1

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

$12 y^{2} y' + 5 \frac{d}{d x} [y^{2}]$

Step 3.3.2

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 3.3.2.1

To apply the Chain Rule, set $u_{2}$ as $y$ .

$12 y^{2} y' + 5 (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [y])$

Step 3.3.2.2

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

$12 y^{2} y' + 5 (2 u_{2} \frac{d}{d x} [y])$

Step 3.3.2.3

Replace all occurrences of $u_{2}$ with $y$ .

$12 y^{2} y' + 5 (2 y \frac{d}{d x} [y])$

Step 3.3.3

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

$12 y^{2} y' + 5 (2 y y')$

Step 3.3.4

Multiply $2$ by $5$ .

$12 y^{2} y' + 10 y y'$

Step 4

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

$2 x = 12 y^{2} y' + 10 y y'$

Step 5

Solve for $y'$ .

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Step 5.1

Rewrite the equation as $12 y^{2} y' + 10 y y' = 2 x$ .

$12 y^{2} y' + 10 y y' = 2 x$

Step 5.2

Factor $2 y y'$ out of $12 y^{2} y' + 10 y y'$ .

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Step 5.2.1

Factor $2 y y'$ out of $12 y^{2} y'$ .

$2 y y' (6 y) + 10 y y' = 2 x$

Step 5.2.2

Factor $2 y y'$ out of $10 y y'$ .

$2 y y' (6 y) + 2 y y' \cdot 5 = 2 x$

Step 5.2.3

Factor $2 y y'$ out of $2 y y' (6 y) + 2 y y' \cdot 5$ .

$2 y y' (6 y + 5) = 2 x$

Step 5.3

Divide each term in $2 y y' (6 y + 5) = 2 x$ by $2 y (6 y + 5)$ and simplify.

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Step 5.3.1

Divide each term in $2 y y' (6 y + 5) = 2 x$ by $2 y (6 y + 5)$ .

$\frac{2 y y' (6 y + 5)}{2 y (6 y + 5)} = \frac{2 x}{2 y (6 y + 5)}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $2$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{2 y y' (6 y + 5)}{2 y (6 y + 5)} = \frac{2 x}{2 y (6 y + 5)}$

Step 5.3.2.1.2

Rewrite the expression.

$\frac{y y' (6 y + 5)}{y (6 y + 5)} = \frac{2 x}{2 y (6 y + 5)}$

Step 5.3.2.2

Cancel the common factor of $y$ .

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Step 5.3.2.2.1

Cancel the common factor.

$\frac{y y' (6 y + 5)}{y (6 y + 5)} = \frac{2 x}{2 y (6 y + 5)}$

Step 5.3.2.2.2

Rewrite the expression.

$\frac{y' (6 y + 5)}{6 y + 5} = \frac{2 x}{2 y (6 y + 5)}$

Step 5.3.2.3

Cancel the common factor of $6 y + 5$ .

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Step 5.3.2.3.1

Cancel the common factor.

$\frac{y' (6 y + 5)}{6 y + 5} = \frac{2 x}{2 y (6 y + 5)}$

Step 5.3.2.3.2

Divide $y'$ by $1$ .

$y' = \frac{2 x}{2 y (6 y + 5)}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Cancel the common factor of $2$ .

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Step 5.3.3.1.1

Cancel the common factor.

$y' = \frac{2 x}{2 y (6 y + 5)}$

Step 5.3.3.1.2

Rewrite the expression.

$y' = \frac{x}{y (6 y + 5)}$

Step 6

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

$\frac{d y}{d x} = \frac{x}{y (6 y + 5)}$