Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Multiply $3$ by $1$ .

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

Step 2.3

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

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Step 2.3.1

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 2.3.1.1

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

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

Step 2.3.1.2

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

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

Step 2.3.1.3

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

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

Step 2.3.2

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

$3 + \frac{1}{y} y'$

Step 2.3.3

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

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

Step 2.4

Reorder terms.

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

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 $f (x) = x^{2}$ and $g (x) = y^{5}$ .

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

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{5}$ and $g (x) = y$ .

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Step 3.2.1

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

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

Step 3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 5$ .

$x^{2} (5 u^{4} \frac{d}{d x} [y]) + y^{5} \frac{d}{d x} [x^{2}]$

Step 3.2.3

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

$x^{2} (5 y^{4} \frac{d}{d x} [y]) + y^{5} \frac{d}{d x} [x^{2}]$

Step 3.3

Move $5$ to the left of $x^{2}$ .

$5 \cdot x^{2} y^{4} \frac{d}{d x} [y] + y^{5} \frac{d}{d x} [x^{2}]$

Step 3.4

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

$5 x^{2} y^{4} y' + y^{5} \frac{d}{d x} [x^{2}]$

Step 3.5

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

$5 x^{2} y^{4} y' + y^{5} (2 x)$

Step 3.6

Reorder.

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Step 3.6.1

Move $2$ to the left of $y^{5}$ .

$5 x^{2} y^{4} y' + 2 \cdot y^{5} x$

Step 3.6.2

Reorder terms.

$5 y^{4} x^{2} y' + 2 y^{5} x$

Step 4

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

$\frac{y'}{y} + 3 = 5 y^{4} x^{2} y' + 2 y^{5} x$

Step 5

Solve for $y'$ .

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Step 5.1

Subtract $5 y^{4} x^{2} y'$ from both sides of the equation.

$\frac{y'}{y} + 3 - 5 y^{4} x^{2} y' = 2 y^{5} x$

Step 5.2

Find the LCD of the terms in the equation.

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Step 5.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y, 1, 1, 1$

Step 5.2.2

The LCM of one and any expression is the expression.

$y$

Step 5.3

Multiply each term in $\frac{y'}{y} + 3 - 5 y^{4} x^{2} y' = 2 y^{5} x$ by $y$ to eliminate the fractions.

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Step 5.3.1

Multiply each term in $\frac{y'}{y} + 3 - 5 y^{4} x^{2} y' = 2 y^{5} x$ by $y$ .

$\frac{y'}{y} y + 3 y - 5 y^{4} x^{2} y' y = 2 y^{5} x y$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Simplify each term.

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Step 5.3.2.1.1

Cancel the common factor of $y$ .

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Step 5.3.2.1.1.1

Cancel the common factor.

$\frac{y'}{y} y + 3 y - 5 y^{4} x^{2} y' y = 2 y^{5} x y$

Step 5.3.2.1.1.2

Rewrite the expression.

$y' + 3 y - 5 y^{4} x^{2} y' y = 2 y^{5} x y$

Step 5.3.2.1.2

Multiply $y^{4}$ by $y$ by adding the exponents.

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Step 5.3.2.1.2.1

Move $y$ .

$y' + 3 y - 5 (y \cdot y^{4}) x^{2} y' = 2 y^{5} x y$

Step 5.3.2.1.2.2

Multiply $y$ by $y^{4}$ .

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Step 5.3.2.1.2.2.1

Raise $y$ to the power of $1$ .

$y' + 3 y - 5 (y^{1} y^{4}) x^{2} y' = 2 y^{5} x y$

Step 5.3.2.1.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y' + 3 y - 5 y^{1 + 4} x^{2} y' = 2 y^{5} x y$

Step 5.3.2.1.2.3

Add $1$ and $4$ .

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

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Multiply $y^{5}$ by $y$ by adding the exponents.

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Step 5.3.3.1.1

Move $y$ .

$y' + 3 y - 5 y^{5} x^{2} y' = 2 (y \cdot y^{5}) x$

Step 5.3.3.1.2

Multiply $y$ by $y^{5}$ .

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Step 5.3.3.1.2.1

Raise $y$ to the power of $1$ .

$y' + 3 y - 5 y^{5} x^{2} y' = 2 (y^{1} y^{5}) x$

Step 5.3.3.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y' + 3 y - 5 y^{5} x^{2} y' = 2 y^{1 + 5} x$

Step 5.3.3.1.3

Add $1$ and $5$ .

$y' + 3 y - 5 y^{5} x^{2} y' = 2 y^{6} x$

Step 5.4

Solve the equation.

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Step 5.4.1

Subtract $3 y$ from both sides of the equation.

$y' - 5 y^{5} x^{2} y' = 2 y^{6} x - 3 y$

Step 5.4.2

Factor $y'$ out of $y' - 5 y^{5} x^{2} y'$ .

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Step 5.4.2.1

Factor $y'$ out of ${y'}^{1}$ .

$y' \cdot 1 - 5 y^{5} x^{2} y' = 2 y^{6} x - 3 y$

Step 5.4.2.2

Factor $y'$ out of $- 5 y^{5} x^{2} y'$ .

$y' \cdot 1 + y' (- 5 y^{5} x^{2}) = 2 y^{6} x - 3 y$

Step 5.4.2.3

Factor $y'$ out of $y' \cdot 1 + y' (- 5 y^{5} x^{2})$ .

$y' (1 - 5 y^{5} x^{2}) = 2 y^{6} x - 3 y$

Step 5.4.3

Divide each term in $y' (1 - 5 y^{5} x^{2}) = 2 y^{6} x - 3 y$ by $1 - 5 y^{5} x^{2}$ and simplify.

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Step 5.4.3.1

Divide each term in $y' (1 - 5 y^{5} x^{2}) = 2 y^{6} x - 3 y$ by $1 - 5 y^{5} x^{2}$ .

$\frac{y' (1 - 5 y^{5} x^{2})}{1 - 5 y^{5} x^{2}} = \frac{2 y^{6} x}{1 - 5 y^{5} x^{2}} + \frac{- 3 y}{1 - 5 y^{5} x^{2}}$

Step 5.4.3.2

Simplify the left side.

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Step 5.4.3.2.1

Cancel the common factor of $1 - 5 y^{5} x^{2}$ .

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Step 5.4.3.2.1.1

Cancel the common factor.

$\frac{y' (1 - 5 y^{5} x^{2})}{1 - 5 y^{5} x^{2}} = \frac{2 y^{6} x}{1 - 5 y^{5} x^{2}} + \frac{- 3 y}{1 - 5 y^{5} x^{2}}$

Step 5.4.3.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{2 y^{6} x}{1 - 5 y^{5} x^{2}} + \frac{- 3 y}{1 - 5 y^{5} x^{2}}$

Step 5.4.3.3

Simplify the right side.

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Step 5.4.3.3.1

Combine the numerators over the common denominator.

$y' = \frac{2 y^{6} x - 3 y}{1 - 5 y^{5} x^{2}}$

Step 5.4.3.3.2

Factor $y$ out of $2 y^{6} x - 3 y$ .

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Step 5.4.3.3.2.1

Factor $y$ out of $2 y^{6} x$ .

$y' = \frac{y (2 y^{5} x) - 3 y}{1 - 5 y^{5} x^{2}}$

Step 5.4.3.3.2.2

Factor $y$ out of $- 3 y$ .

$y' = \frac{y (2 y^{5} x) + y \cdot - 3}{1 - 5 y^{5} x^{2}}$

Step 5.4.3.3.2.3

Factor $y$ out of $y (2 y^{5} x) + y \cdot - 3$ .

$y' = \frac{y (2 y^{5} x - 3)}{1 - 5 y^{5} x^{2}}$

Step 6

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

$\frac{d y}{d x} = \frac{y (2 y^{5} x - 3)}{1 - 5 y^{5} x^{2}}$