Calculus Examples

$f (x) = x^{6 x}$

Step 1

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

$\ln (f (x)) = \ln (x^{6 x})$

Step 2

Expand $\ln (x^{6 x})$ by moving $6 x$ outside the logarithm.

$\ln (f (x)) = 6 x \ln (x)$

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 $\ln (f (x))$ using the chain rule.

$\frac{x'}{x} = 6 x \ln (x)$

Step 3.2

Differentiate the right hand side.

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Step 3.2.1

Differentiate $6 x \ln (x)$ .

$\frac{x'}{x} = \frac{d}{d x} [6 x \ln (x)]$

Step 3.2.2

Since $6$ is constant with respect to $x$ , the derivative of $6 x \ln (x)$ with respect to $x$ is $6 \frac{d}{d x} [x \ln (x)]$ .

$\frac{x'}{x} = 6 \frac{d}{d x} [x \ln (x)]$

Step 3.2.3

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 $g (x) = \ln (x)$ .

$\frac{x'}{x} = 6 (x \frac{d}{d x} [\ln (x)] + \ln (x) \frac{d}{d x} [x])$

Step 3.2.4

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

$\frac{x'}{x} = 6 (x \frac{1}{x} + \ln (x) \frac{d}{d x} [x])$

Step 3.2.5

Differentiate using the Power Rule.

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Step 3.2.5.1

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

$\frac{x'}{x} = 6 (\frac{x}{x} + \ln (x) \frac{d}{d x} [x])$

Step 3.2.5.2

Cancel the common factor of $x$ .

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Step 3.2.5.2.1

Cancel the common factor.

$\frac{x'}{x} = 6 (\frac{x}{x} + \ln (x) \frac{d}{d x} [x])$

Step 3.2.5.2.2

Rewrite the expression.

$\frac{x'}{x} = 6 (1 + \ln (x) \frac{d}{d x} [x])$

Step 3.2.5.3

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

$\frac{x'}{x} = 6 (1 + \ln (x) \cdot 1)$

Step 3.2.5.4

Multiply $\ln (x)$ by $1$ .

$\frac{x'}{x} = 6 (1 + \ln (x))$

Step 3.2.6

Simplify.

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Step 3.2.6.1

Apply the distributive property.

$\frac{x'}{x} = 6 \cdot 1 + 6 \ln (x)$

Step 3.2.6.2

Multiply $6$ by $1$ .

$\frac{x'}{x} = 6 + 6 \ln (x)$

Step 3.2.6.3

Reorder terms.

$\frac{x'}{x} = 6 \ln (x) + 6$

Step 4

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

$x' = (6 \ln (x) + 6) x^{6 x}$

Step 5

Simplify the right hand side.

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Step 5.1

Simplify $6 \ln (x)$ by moving $6$ inside the logarithm.

$x' = (\ln (x^{6}) + 6) x^{6 x}$

Step 5.2

Apply the distributive property.

$x' = \ln (x^{6}) x^{6 x} + 6 x^{6 x}$

Step 5.3

Reorder factors in $\ln (x^{6}) x^{6 x} + 6 x^{6 x}$ .

$x' = x^{6 x} \ln (x^{6}) + 6 x^{6 x}$