Calculus Examples

$y = x^{3 x}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (x^{3 x})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Use the properties of logarithms to simplify the differentiation.

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Step 3.1.1

Rewrite $x^{3 x}$ as $e^{\ln (x^{3 x})}$ .

$\frac{d}{d x} [e^{\ln (x^{3 x})}]$

Step 3.1.2

Expand $\ln (x^{3 x})$ by moving $3 x$ outside the logarithm.

$\frac{d}{d x} [e^{3 x \ln (x)}]$

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 $f (x) = e^{x}$ and $g (x) = 3 x \ln (x)$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $3 x \ln (x)$ .

$\frac{d}{d u} [e^{u}] \frac{d}{d x} [3 x \ln (x)]$

Step 3.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$e^{u} \frac{d}{d x} [3 x \ln (x)]$

Step 3.2.3

Replace all occurrences of $u$ with $3 x \ln (x)$ .

$e^{3 x \ln (x)} \frac{d}{d x} [3 x \ln (x)]$

Step 3.3

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

$e^{3 x \ln (x)} (3 \frac{d}{d x} [x \ln (x)])$

Step 3.4

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

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

Step 3.5

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

$e^{3 x \ln (x)} (3 (x \frac{1}{x} + \ln (x) \frac{d}{d x} [x]))$

Step 3.6

Differentiate using the Power Rule.

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Step 3.6.1

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

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

Step 3.6.2

Cancel the common factor of $x$ .

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Step 3.6.2.1

Cancel the common factor.

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

Step 3.6.2.2

Rewrite the expression.

$e^{3 x \ln (x)} (3 (1 + \ln (x) \frac{d}{d x} [x]))$

Step 3.6.3

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

$e^{3 x \ln (x)} (3 (1 + \ln (x) \cdot 1))$

Step 3.6.4

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

$e^{3 x \ln (x)} (3 (1 + \ln (x)))$

Step 3.7

Simplify.

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Step 3.7.1

Apply the distributive property.

$e^{3 x \ln (x)} (3 \cdot 1 + 3 \ln (x))$

Step 3.7.2

Apply the distributive property.

$e^{3 x \ln (x)} (3 \cdot 1) + e^{3 x \ln (x)} (3 \ln (x))$

Step 3.7.3

Combine terms.

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Step 3.7.3.1

Multiply $3$ by $1$ .

$e^{3 x \ln (x)} \cdot 3 + e^{3 x \ln (x)} (3 \ln (x))$

Step 3.7.3.2

Move $3$ to the left of $e^{3 x \ln (x)}$ .

$3 e^{3 x \ln (x)} + e^{3 x \ln (x)} (3 \ln (x))$

Step 3.7.4

Reorder terms.

$3 e^{3 x \ln (x)} \ln (x) + 3 e^{3 x \ln (x)}$

Step 4

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

$y' = 3 e^{3 x \ln (x)} \ln (x) + 3 e^{3 x \ln (x)}$

Step 5

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

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