Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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 2.2.2.1

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

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

Step 2.2.2.2

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

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

Step 2.2.2.3

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

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

Step 2.2.3

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

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

Step 2.2.4

Multiply $2$ by $5$ .

$10 y y' + \frac{d}{d x} [5] + \frac{d}{d x} [x^{3}]$

Step 2.3

Differentiate.

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Step 2.3.1

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$10 y y' + 0 + \frac{d}{d x} [x^{3}]$

Step 2.3.2

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

$10 y y' + 0 + 3 x^{2}$

Step 2.4

Simplify.

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Step 2.4.1

Add $10 y y'$ and $0$ .

$10 y y' + 3 x^{2}$

Step 2.4.2

Reorder terms.

$3 x^{2} + 10 y y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.2

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

$3 y^{2} y'$

Step 4

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

$3 x^{2} + 10 y y' = 3 y^{2} y'$

Step 5

Solve for $y'$ .

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Step 5.1

Subtract $3 y^{2} y'$ from both sides of the equation.

$3 x^{2} + 10 y y' - 3 y^{2} y' = 0$

Step 5.2

Subtract $3 x^{2}$ from both sides of the equation.

$10 y y' - 3 y^{2} y' = - 3 x^{2}$

Step 5.3

Factor $y y'$ out of $10 y y' - 3 y^{2} y'$ .

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Step 5.3.1

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

$y y' \cdot 10 - 3 y^{2} y' = - 3 x^{2}$

Step 5.3.2

Factor $y y'$ out of $- 3 y^{2} y'$ .

$y y' \cdot 10 + y y' (- 3 y) = - 3 x^{2}$

Step 5.3.3

Factor $y y'$ out of $y y' \cdot 10 + y y' (- 3 y)$ .

$y y' (10 - 3 y) = - 3 x^{2}$

Step 5.4

Divide each term in $y y' (10 - 3 y) = - 3 x^{2}$ by $y (10 - 3 y)$ and simplify.

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Step 5.4.1

Divide each term in $y y' (10 - 3 y) = - 3 x^{2}$ by $y (10 - 3 y)$ .

$\frac{y y' (10 - 3 y)}{y (10 - 3 y)} = \frac{- 3 x^{2}}{y (10 - 3 y)}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $y$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{y y' (10 - 3 y)}{y (10 - 3 y)} = \frac{- 3 x^{2}}{y (10 - 3 y)}$

Step 5.4.2.1.2

Rewrite the expression.

$\frac{y' (10 - 3 y)}{10 - 3 y} = \frac{- 3 x^{2}}{y (10 - 3 y)}$

Step 5.4.2.2

Cancel the common factor of $10 - 3 y$ .

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Step 5.4.2.2.1

Cancel the common factor.

$\frac{y' (10 - 3 y)}{10 - 3 y} = \frac{- 3 x^{2}}{y (10 - 3 y)}$

Step 5.4.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{- 3 x^{2}}{y (10 - 3 y)}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Move the negative in front of the fraction.

$y' = - \frac{3 x^{2}}{y (10 - 3 y)}$

Step 6

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

$\frac{d y}{d x} = - \frac{3 x^{2}}{y (10 - 3 y)}$