Calculus Examples

$x^{3} - y = \ln (y)$

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

Differentiate.

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

$3 x^{2} - y'$

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 $f (x) = \ln (x)$ and $g (x) = y$ .

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Step 3.1.1

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

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [y]$

Step 3.1.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$\frac{1}{u} \frac{d}{d x} [y]$

Step 3.1.3

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

$\frac{1}{y} \frac{d}{d x} [y]$

Step 3.2

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

$\frac{1}{y} y'$

Step 3.3

Combine $\frac{1}{y}$ and $y'$ .

$\frac{y'}{y}$

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

Multiply both sides by $y$ .

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

Step 5.2

Simplify.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Apply the distributive property.

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

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Cancel the common factor of $y$ .

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Step 5.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.1.2

Rewrite the expression.

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

Step 5.3

Solve for $y'$ .

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Step 5.3.1

Subtract $y'$ from both sides of the equation.

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

Step 5.3.2

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

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

Step 5.3.3

Factor $y'$ out of $- y' y - y'$ .

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Step 5.3.3.1

Factor $y'$ out of $- y' y$ .

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

Step 5.3.3.2

Factor $y'$ out of $- y'$ .

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

Step 5.3.3.3

Factor $y'$ out of $y' (- 1 y) + y' \cdot - 1$ .

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

Step 5.3.4

Rewrite $- 1 y$ as $- y$ .

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

Step 5.3.5

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

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Step 5.3.5.1

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

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

Step 5.3.5.2

Simplify the left side.

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Step 5.3.5.2.1

Cancel the common factor of $- y - 1$ .

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Step 5.3.5.2.1.1

Cancel the common factor.

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

Step 5.3.5.2.1.2

Divide $y'$ by $1$ .

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

Step 5.3.5.3

Simplify the right side.

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Step 5.3.5.3.1

Move the negative in front of the fraction.

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

Step 5.3.5.3.2

Factor $- 1$ out of $- y$ .

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

Step 5.3.5.3.3

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 5.3.5.3.4

Factor $- 1$ out of $- (y) - 1 (1)$ .

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

Step 5.3.5.3.5

Simplify the expression.

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Step 5.3.5.3.5.1

Rewrite $- (y + 1)$ as $- 1 (y + 1)$ .

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

Step 5.3.5.3.5.2

Move the negative in front of the fraction.

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

Step 5.3.5.3.5.3

Multiply $- 1$ by $- 1$ .

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

Step 5.3.5.3.5.4

Multiply $\frac{3 x^{2} y}{y + 1}$ by $1$ .

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

Step 6

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

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