Calculus Examples

$y = x^{x^{2}}$

Step 1

Let $y = f (x)$ , take the natural logarithm of both sides $\ln (y) = \ln (f (x))$ .

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

Step 2

Expand $\ln (x^{x^{2}})$ by moving $x^{2}$ outside the logarithm.

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

Step 3

Differentiate the expression using the chain rule, keeping in mind that $y$ is a function of $x$ .

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Step 3.1

Differentiate the left hand side $\ln (y)$ using the chain rule.

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

Step 3.2

Differentiate the right hand side.

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Step 3.2.1

Differentiate $x^{2} \ln (x)$ .

$\frac{y'}{y} = \frac{d}{d x} [x^{2} \ln (x)]$

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

Differentiate using the Power Rule.

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Step 3.2.4.1

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

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

Step 3.2.4.2

Cancel the common factor of $x^{2}$ and $x$ .

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Step 3.2.4.2.1

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

$\frac{y'}{y} = \frac{x \cdot x}{x} + \ln (x) \frac{d}{d x} [x^{2}]$

Step 3.2.4.2.2

Cancel the common factors.

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Step 3.2.4.2.2.1

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

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

Step 3.2.4.2.2.2

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

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

Step 3.2.4.2.2.3

Cancel the common factor.

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

Step 3.2.4.2.2.4

Rewrite the expression.

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

Step 3.2.4.2.2.5

Divide $x$ by $1$ .

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

Step 3.2.4.3

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

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

Step 3.2.4.4

Reorder terms.

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

Step 4

Isolate $y'$ and substitute the original function for $y$ in the right hand side.

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

Step 5

Simplify the right hand side.

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Step 5.1

Simplify $2 \ln (x)$ by moving $2$ inside the logarithm.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Multiply $x$ by $x^{x^{2}}$ by adding the exponents.

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Step 5.3.1

Move $x^{x^{2}}$ .

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

Step 5.3.2

Multiply $x^{x^{2}}$ by $x$ .

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Step 5.3.2.1

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

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

Step 5.3.2.2

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

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

Step 5.4

Multiply $x$ by $x^{x^{2}}$ .

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Step 5.4.1

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

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

Step 5.4.2

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

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