Calculus Examples

$3 x^{2} + 5 \ln (y) = 12$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (3 x^{2} + 5 \ln (y)) = \frac{d}{d x} (12)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

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

Step 2.2.2

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

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

Step 2.2.3

Multiply $2$ by $3$ .

$6 x + \frac{d}{d x} [5 \ln (y)]$

Step 2.3

Evaluate $\frac{d}{d x} [5 \ln (y)]$ .

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Step 2.3.1

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

$6 x + 5 \frac{d}{d x} [\ln (y)]$

Step 2.3.2

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

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Step 2.3.2.1

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

$6 x + 5 (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [y])$

Step 2.3.2.2

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

$6 x + 5 (\frac{1}{u} \frac{d}{d x} [y])$

Step 2.3.2.3

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

$6 x + 5 (\frac{1}{y} \frac{d}{d x} [y])$

Step 2.3.3

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

$6 x + 5 (\frac{1}{y} y')$

Step 2.3.4

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

$6 x + 5 \frac{y'}{y}$

Step 2.3.5

Combine $5$ and $\frac{y'}{y}$ .

$6 x + \frac{5 y'}{y}$

Step 3

Since $12$ is constant with respect to $x$ , the derivative of $12$ with respect to $x$ is $0$ .

$0$

Step 4

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

$6 x + \frac{5 y'}{y} = 0$

Step 5

Solve for $y'$ .

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Step 5.1

Subtract $6 x$ from both sides of the equation.

$\frac{5 y'}{y} = - 6 x$

Step 5.2

Multiply both sides by $y$ .

$\frac{5 y'}{y} y = - 6 x y$

Step 5.3

Simplify the left side.

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Step 5.3.1

Cancel the common factor of $y$ .

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Step 5.3.1.1

Cancel the common factor.

$\frac{5 y'}{y} y = - 6 x y$

Step 5.3.1.2

Rewrite the expression.

$5 y' = - 6 x y$

Step 5.4

Divide each term in $5 y' = - 6 x y$ by $5$ and simplify.

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Step 5.4.1

Divide each term in $5 y' = - 6 x y$ by $5$ .

$\frac{5 y'}{5} = \frac{- 6 x y}{5}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $5$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{5 y'}{5} = \frac{- 6 x y}{5}$

Step 5.4.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{- 6 x y}{5}$

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.

$y' = - \frac{6 x y}{5}$

Step 6

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

$\frac{d y}{d x} = - \frac{6 x y}{5}$