Calculus Examples

$\ln (y) = 7 x \ln (\sqrt{x})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\ln (y) = 7 x \ln (x^{\frac{1}{2}})$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (\ln (y)) = \frac{d}{d x} (7 x \ln (x^{\frac{1}{2}}))$

Step 3

Differentiate the left 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

Differentiate the right side of the equation.

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Step 4.1

Since $7$ is constant with respect to $x$ , the derivative of $7 x \ln (x^{\frac{1}{2}})$ with respect to $x$ is $7 \frac{d}{d x} [x \ln (x^{\frac{1}{2}})]$ .

$7 \frac{d}{d x} [x \ln (x^{\frac{1}{2}})]$

Step 4.2

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 $f (x) = x$ and $g (x) = \ln (x^{\frac{1}{2}})$ .

$7 (x \frac{d}{d x} [\ln (x^{\frac{1}{2}})] + \ln (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 4.3

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 $g (x) = x^{\frac{1}{2}}$ .

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Step 4.3.1

To apply the Chain Rule, set $u$ as $x^{\frac{1}{2}}$ .

$7 (x (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [x^{\frac{1}{2}}]) + \ln (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 4.3.2

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

$7 (x (\frac{1}{u} \frac{d}{d x} [x^{\frac{1}{2}}]) + \ln (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 4.3.3

Replace all occurrences of $u$ with $x^{\frac{1}{2}}$ .

$7 (x (\frac{1}{x^{\frac{1}{2}}} \frac{d}{d x} [x^{\frac{1}{2}}]) + \ln (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 4.4

Combine $\frac{1}{x^{\frac{1}{2}}}$ and $x$ .

$7 (\frac{x}{x^{\frac{1}{2}}} \frac{d}{d x} [x^{\frac{1}{2}}] + \ln (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 4.5

Move $x^{\frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$7 (x \cdot x^{- \frac{1}{2}} \frac{d}{d x} [x^{\frac{1}{2}}] + \ln (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 4.6

Multiply $x$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 4.6.1

Multiply $x$ by $x^{- \frac{1}{2}}$ .

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Step 4.6.1.1

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

$7 (x^{1} x^{- \frac{1}{2}} \frac{d}{d x} [x^{\frac{1}{2}}] + \ln (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 4.6.1.2

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

$7 (x^{1 - \frac{1}{2}} \frac{d}{d x} [x^{\frac{1}{2}}] + \ln (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 4.6.2

Write $1$ as a fraction with a common denominator.

$7 (x^{\frac{2}{2} - \frac{1}{2}} \frac{d}{d x} [x^{\frac{1}{2}}] + \ln (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 4.6.3

Combine the numerators over the common denominator.

$7 (x^{\frac{2 - 1}{2}} \frac{d}{d x} [x^{\frac{1}{2}}] + \ln (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 4.6.4

Subtract $1$ from $2$ .

$7 (x^{\frac{1}{2}} \frac{d}{d x} [x^{\frac{1}{2}}] + \ln (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 4.7

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

$7 (x^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1}{2} - 1}) + \ln (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 4.8

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$7 (x^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) + \ln (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 4.9

Combine $- 1$ and $\frac{2}{2}$ .

$7 (x^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + \ln (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 4.10

Combine the numerators over the common denominator.

$7 (x^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}) + \ln (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 4.11

Simplify the numerator.

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Step 4.11.1

Multiply $- 1$ by $2$ .

$7 (x^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1 - 2}{2}}) + \ln (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 4.11.2

Subtract $2$ from $1$ .

$7 (x^{\frac{1}{2}} (\frac{1}{2} x^{\frac{- 1}{2}}) + \ln (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 4.12

Move the negative in front of the fraction.

$7 (x^{\frac{1}{2}} (\frac{1}{2} x^{- \frac{1}{2}}) + \ln (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 4.13

Combine $\frac{1}{2}$ and $x^{- \frac{1}{2}}$ .

$7 (x^{\frac{1}{2}} \frac{x^{- \frac{1}{2}}}{2} + \ln (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 4.14

Combine $x^{\frac{1}{2}}$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$7 (\frac{x^{\frac{1}{2}} x^{- \frac{1}{2}}}{2} + \ln (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 4.15

Multiply $x^{\frac{1}{2}}$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 4.15.1

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

$7 (\frac{x^{\frac{1}{2} - \frac{1}{2}}}{2} + \ln (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 4.15.2

Combine the numerators over the common denominator.

$7 (\frac{x^{\frac{1 - 1}{2}}}{2} + \ln (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 4.15.3

Subtract $1$ from $1$ .

$7 (\frac{x^{\frac{0}{2}}}{2} + \ln (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 4.15.4

Divide $0$ by $2$ .

$7 (\frac{x^{0}}{2} + \ln (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 4.16

Simplify $x^{0}$ .

$7 (\frac{1}{2} + \ln (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 4.17

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

$7 (\frac{1}{2} + \ln (x^{\frac{1}{2}}) \cdot 1)$

Step 4.18

Multiply $\ln (x^{\frac{1}{2}})$ by $1$ .

$7 (\frac{1}{2} + \ln (x^{\frac{1}{2}}))$

Step 4.19

Simplify.

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Step 4.19.1

Apply the distributive property.

$7 (\frac{1}{2}) + 7 \ln (x^{\frac{1}{2}})$

Step 4.19.2

Combine $7$ and $\frac{1}{2}$ .

$\frac{7}{2} + 7 \ln (x^{\frac{1}{2}})$

Step 4.19.3

Reorder terms.

$7 \ln (x^{\frac{1}{2}}) + \frac{7}{2}$

Step 5

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

$\frac{y'}{y} = 7 \ln (x^{\frac{1}{2}}) + \frac{7}{2}$

Step 6

Solve for $y'$ .

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Step 6.1

Multiply both sides by $y$ .

$\frac{y'}{y} y = (7 \ln (x^{\frac{1}{2}}) + \frac{7}{2}) y$

Step 6.2

Simplify.

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Step 6.2.1

Simplify the left side.

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Step 6.2.1.1

Cancel the common factor of $y$ .

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Step 6.2.1.1.1

Cancel the common factor.

$\frac{y'}{y} y = (7 \ln (x^{\frac{1}{2}}) + \frac{7}{2}) y$

Step 6.2.1.1.2

Rewrite the expression.

$y' = (7 \ln (x^{\frac{1}{2}}) + \frac{7}{2}) y$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Simplify $(7 \ln (x^{\frac{1}{2}}) + \frac{7}{2}) y$ .

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Step 6.2.2.1.1

Simplify each term.

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Step 6.2.2.1.1.1

Simplify $7 \ln (x^{\frac{1}{2}})$ by moving $7$ inside the logarithm.

$y' = (\ln ({(x^{\frac{1}{2}})}^{7}) + \frac{7}{2}) y$

Step 6.2.2.1.1.2

Multiply the exponents in ${(x^{\frac{1}{2}})}^{7}$ .

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Step 6.2.2.1.1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y' = (\ln (x^{\frac{1}{2} \cdot 7}) + \frac{7}{2}) y$

Step 6.2.2.1.1.2.2

Combine $\frac{1}{2}$ and $7$ .

$y' = (\ln (x^{\frac{7}{2}}) + \frac{7}{2}) y$

Step 6.2.2.1.2

Simplify terms.

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Step 6.2.2.1.2.1

Apply the distributive property.

$y' = \ln (x^{\frac{7}{2}}) y + \frac{7}{2} y$

Step 6.2.2.1.2.2

Combine $\frac{7}{2}$ and $y$ .

$y' = \ln (x^{\frac{7}{2}}) y + \frac{7 y}{2}$

Step 6.2.2.1.2.3

Reorder factors in $\ln (x^{\frac{7}{2}}) y + \frac{7 y}{2}$ .

$y' = y \ln (x^{\frac{7}{2}}) + \frac{7 y}{2}$

Step 7

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

$\frac{d y}{d x} = y \ln (x^{\frac{7}{2}}) + \frac{7 y}{2}$