Calculus Examples

$y = x \ln (x)$

Step 1

Differentiate both sides of the equation.

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

Step 2

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

$1$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the Product Rule which states that $\frac{d}{d y} [f (y) g (y)]$ is $f (y) \frac{d}{d y} [g (y)] + g (y) \frac{d}{d y} [f (y)]$ where $f (y) = x$ and $g (y) = \ln (x)$ .

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.3

Simplify terms.

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Step 3.3.1

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

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

Step 3.3.2

Cancel the common factor of $x$ .

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Step 3.3.2.1

Cancel the common factor.

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

Step 3.3.2.2

Rewrite the expression.

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

Step 3.3.3

Multiply $\frac{d}{d y} [x]$ by $1$ .

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

Step 3.4

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

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

Step 3.5

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

$x' + \ln (x) x'$

Step 3.6

Reorder terms.

$\ln (x) x' + x'$

Step 4

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

$1 = \ln (x) x' + x'$

Step 5

Solve for $x'$ .

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Step 5.1

Rewrite the equation as $\ln (x) x' + x' = 1$ .

$\ln (x) x' + x' = 1$

Step 5.2

Reorder factors in $\ln (x) x' + x'$ .

$x' \ln (x) + x' = 1$

Step 5.3

Factor $x'$ out of $x' \ln (x) + x'$ .

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Step 5.3.1

Factor $x'$ out of $x' \ln (x)$ .

$x' (\ln (x)) + x' = 1$

Step 5.3.2

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

$x' (\ln (x)) + x' = 1$

Step 5.3.3

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

$x' (\ln (x)) + x' \cdot 1 = 1$

Step 5.3.4

Factor $x'$ out of $x' (\ln (x)) + x' \cdot 1$ .

$x' (\ln (x) + 1) = 1$

Step 5.4

Divide each term in $x' (\ln (x) + 1) = 1$ by $\ln (x) + 1$ and simplify.

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Step 5.4.1

Divide each term in $x' (\ln (x) + 1) = 1$ by $\ln (x) + 1$ .

$\frac{x' (\ln (x) + 1)}{\ln (x) + 1} = \frac{1}{\ln (x) + 1}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $\ln (x) + 1$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{x' (\ln (x) + 1)}{\ln (x) + 1} = \frac{1}{\ln (x) + 1}$

Step 5.4.2.1.2

Divide $x'$ by $1$ .

$x' = \frac{1}{\ln (x) + 1}$

Step 6

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

$\frac{d x}{d y} = \frac{1}{\ln (x) + 1}$