Calculus Examples

$3 y^{3} + 1 + x = x^{3} y$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (3 y^{3} + 1 + x) = \frac{d}{d x} (x^{3} y)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $3 y^{3} + 1 + x$ with respect to $x$ is $\frac{d}{d x} [3 y^{3}] + \frac{d}{d x} [1] + \frac{d}{d x} [x]$ .

$\frac{d}{d x} [3 y^{3}] + \frac{d}{d x} [1] + \frac{d}{d x} [x]$

Step 2.2

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

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Step 2.2.1

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

$3 \frac{d}{d x} [y^{3}] + \frac{d}{d x} [1] + \frac{d}{d x} [x]$

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 $f (x) = x^{3}$ and $g (x) = y$ .

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Step 2.2.2.1

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

$3 (\frac{d}{d u} [u^{3}] \frac{d}{d x} [y]) + \frac{d}{d x} [1] + \frac{d}{d x} [x]$

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 $n = 3$ .

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

Step 2.2.2.3

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

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

Step 2.2.3

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

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

Step 2.2.4

Multiply $3$ by $3$ .

$9 y^{2} y' + \frac{d}{d x} [1] + \frac{d}{d x} [x]$

Step 2.3

Differentiate.

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Step 2.3.1

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$9 y^{2} y' + 0 + \frac{d}{d x} [x]$

Step 2.3.2

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

$9 y^{2} y' + 0 + 1$

Step 2.3.3

Add $9 y^{2} y'$ and $0$ .

$9 y^{2} y' + 1$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Reorder.

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Step 3.4.1

Move $3$ to the left of $y$ .

$x^{3} y' + 3 \cdot y x^{2}$

Step 3.4.2

Reorder terms.

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

Step 4

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

$9 y^{2} y' + 1 = x^{3} y' + 3 x^{2} y$

Step 5

Solve for $y'$ .

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Step 5.1

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

$9 y^{2} y' + 1 - x^{3} y' = 3 x^{2} y$

Step 5.2

Subtract $1$ from both sides of the equation.

$9 y^{2} y' - x^{3} y' = 3 x^{2} y - 1$

Step 5.3

Factor $y'$ out of $9 y^{2} y' - x^{3} y'$ .

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Step 5.3.1

Factor $y'$ out of $9 y^{2} y'$ .

$y' (9 y^{2}) - x^{3} y' = 3 x^{2} y - 1$

Step 5.3.2

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

$y' (9 y^{2}) + y' (- x^{3}) = 3 x^{2} y - 1$

Step 5.3.3

Factor $y'$ out of $y' (9 y^{2}) + y' (- x^{3})$ .

$y' (9 y^{2} - x^{3}) = 3 x^{2} y - 1$

Step 5.4

Divide each term in $y' (9 y^{2} - x^{3}) = 3 x^{2} y - 1$ by $9 y^{2} - x^{3}$ and simplify.

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Step 5.4.1

Divide each term in $y' (9 y^{2} - x^{3}) = 3 x^{2} y - 1$ by $9 y^{2} - x^{3}$ .

$\frac{y' (9 y^{2} - x^{3})}{9 y^{2} - x^{3}} = \frac{3 x^{2} y}{9 y^{2} - x^{3}} + \frac{- 1}{9 y^{2} - x^{3}}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $9 y^{2} - x^{3}$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{y' (9 y^{2} - x^{3})}{9 y^{2} - x^{3}} = \frac{3 x^{2} y}{9 y^{2} - x^{3}} + \frac{- 1}{9 y^{2} - x^{3}}$

Step 5.4.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{3 x^{2} y}{9 y^{2} - x^{3}} + \frac{- 1}{9 y^{2} - x^{3}}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Combine the numerators over the common denominator.

$y' = \frac{3 x^{2} y - 1}{9 y^{2} - x^{3}}$

Step 6

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

$\frac{d y}{d x} = \frac{3 x^{2} y - 1}{9 y^{2} - x^{3}}$