Calculus Examples

$y = \ln (3 x + 4 y)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\ln (3 x + 4 y))$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

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 $f (x) = \ln (x)$ and $g (x) = 3 x + 4 y$ .

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Step 3.1.1

To apply the Chain Rule, set $u$ as $3 x + 4 y$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [3 x + 4 y]$

Step 3.1.2

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

$\frac{1}{u} \frac{d}{d x} [3 x + 4 y]$

Step 3.1.3

Replace all occurrences of $u$ with $3 x + 4 y$ .

$\frac{1}{3 x + 4 y} \frac{d}{d x} [3 x + 4 y]$

Step 3.2

Differentiate.

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

$\frac{1}{3 x + 4 y} (3 \cdot 1 + \frac{d}{d x} [4 y])$

Step 3.2.4

Multiply $3$ by $1$ .

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

Step 3.2.5

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

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

Step 3.3

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

$\frac{1}{3 x + 4 y} (3 + 4 y')$

Step 3.4

Simplify.

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Step 3.4.1

Reorder the factors of $\frac{1}{3 x + 4 y} (3 + 4 y')$ .

$(3 + 4 y') \frac{1}{3 x + 4 y}$

Step 3.4.2

Multiply $3 + 4 y'$ by $\frac{1}{3 x + 4 y}$ .

$\frac{3 + 4 y'}{3 x + 4 y}$

Step 4

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

$y' = \frac{3 + 4 y'}{3 x + 4 y}$

Step 5

Solve for $y'$ .

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Step 5.1

Multiply both sides by $3 x + 4 y$ .

$y' (3 x + 4 y) = \frac{3 + 4 y'}{3 x + 4 y} (3 x + 4 y)$

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

Simplify $y' (3 x + 4 y)$ .

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Step 5.2.1.1.1

Apply the distributive property.

$y' (3 x) + y' (4 y) = \frac{3 + 4 y'}{3 x + 4 y} (3 x + 4 y)$

Step 5.2.1.1.2

Reorder.

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Step 5.2.1.1.2.1

Rewrite using the commutative property of multiplication.

$3 y' x + y' (4 y) = \frac{3 + 4 y'}{3 x + 4 y} (3 x + 4 y)$

Step 5.2.1.1.2.2

Rewrite using the commutative property of multiplication.

$3 y' x + 4 y' y = \frac{3 + 4 y'}{3 x + 4 y} (3 x + 4 y)$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Simplify $\frac{3 + 4 y'}{3 x + 4 y} (3 x + 4 y)$ .

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Step 5.2.2.1.1

Cancel the common factor of $3 x + 4 y$ .

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Step 5.2.2.1.1.1

Cancel the common factor.

$3 y' x + 4 y' y = \frac{3 + 4 y'}{3 x + 4 y} (3 x + 4 y)$

Step 5.2.2.1.1.2

Rewrite the expression.

$3 y' x + 4 y' y = 3 + 4 y'$

Step 5.2.2.1.2

Reorder $3$ and $4 y'$ .

$3 y' x + 4 y' y = 4 y' + 3$

Step 5.3

Solve for $y'$ .

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Step 5.3.1

Subtract $4 y'$ from both sides of the equation.

$3 y' x + 4 y' y - 4 y' = 3$

Step 5.3.2

Factor $y'$ out of $3 y' x + 4 y' y - 4 y'$ .

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Step 5.3.2.1

Factor $y'$ out of $3 y' x$ .

$y' (3 x) + 4 y' y - 4 y' = 3$

Step 5.3.2.2

Factor $y'$ out of $4 y' y$ .

$y' (3 x) + y' (4 y) - 4 y' = 3$

Step 5.3.2.3

Factor $y'$ out of $- 4 y'$ .

$y' (3 x) + y' (4 y) + y' \cdot - 4 = 3$

Step 5.3.2.4

Factor $y'$ out of $y' (3 x) + y' (4 y)$ .

$y' (3 x + 4 y) + y' \cdot - 4 = 3$

Step 5.3.2.5

Factor $y'$ out of $y' (3 x + 4 y) + y' \cdot - 4$ .

$y' (3 x + 4 y - 4) = 3$

Step 5.3.3

Divide each term in $y' (3 x + 4 y - 4) = 3$ by $3 x + 4 y - 4$ and simplify.

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Step 5.3.3.1

Divide each term in $y' (3 x + 4 y - 4) = 3$ by $3 x + 4 y - 4$ .

$\frac{y' (3 x + 4 y - 4)}{3 x + 4 y - 4} = \frac{3}{3 x + 4 y - 4}$

Step 5.3.3.2

Simplify the left side.

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Step 5.3.3.2.1

Cancel the common factor of $3 x + 4 y - 4$ .

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Step 5.3.3.2.1.1

Cancel the common factor.

$\frac{y' (3 x + 4 y - 4)}{3 x + 4 y - 4} = \frac{3}{3 x + 4 y - 4}$

Step 5.3.3.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{3}{3 x + 4 y - 4}$

Step 6

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

$\frac{d y}{d x} = \frac{3}{3 x + 4 y - 4}$