Calculus Examples

$y = \log_{2} (4 x^{3})$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\log_{2} (4 x^{3}))$

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

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \log_{2} (x)$ and $g (x) = 4 x^{3}$ .

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Step 3.1.1

To apply the Chain Rule, set $u$ as $4 x^{3}$ .

$\frac{d}{d u} [\log_{2} (u)] \frac{d}{d x} [4 x^{3}]$

Step 3.1.2

The derivative of $\log_{2} (u)$ with respect to $u$ is $\frac{1}{u \ln (2)}$ .

$\frac{1}{u \ln (2)} \frac{d}{d x} [4 x^{3}]$

Step 3.1.3

Replace all occurrences of $u$ with $4 x^{3}$ .

$\frac{1}{4 x^{3} \ln (2)} \frac{d}{d x} [4 x^{3}]$

Step 3.2

Differentiate.

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Step 3.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{3}$ with respect to $x$ is $4 \frac{d}{d x} [x^{3}]$ .

$\frac{1}{4 x^{3} \ln (2)} (4 \frac{d}{d x} [x^{3}])$

Step 3.2.2

Simplify terms.

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Step 3.2.2.1

Combine $4$ and $\frac{1}{4 x^{3} \ln (2)}$ .

$\frac{4}{4 x^{3} \ln (2)} \frac{d}{d x} [x^{3}]$

Step 3.2.2.2

Cancel the common factor of $4$ .

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Step 3.2.2.2.1

Cancel the common factor.

$\frac{4}{4 x^{3} \ln (2)} \frac{d}{d x} [x^{3}]$

Step 3.2.2.2.2

Rewrite the expression.

$\frac{1}{x^{3} \ln (2)} \frac{d}{d x} [x^{3}]$

Step 3.2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$\frac{1}{x^{3} \ln (2)} (3 x^{2})$

Step 3.2.4

Simplify terms.

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Step 3.2.4.1

Combine $3$ and $\frac{1}{x^{3} \ln (2)}$ .

$\frac{3}{x^{3} \ln (2)} x^{2}$

Step 3.2.4.2

Combine $\frac{3}{x^{3} \ln (2)}$ and $x^{2}$ .

$\frac{3 x^{2}}{x^{3} \ln (2)}$

Step 3.2.4.3

Cancel the common factor of $x^{2}$ and $x^{3}$ .

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Step 3.2.4.3.1

Factor $x^{2}$ out of $3 x^{2}$ .

$\frac{x^{2} \cdot 3}{x^{3} \ln (2)}$

Step 3.2.4.3.2

Cancel the common factors.

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Step 3.2.4.3.2.1

Factor $x^{2}$ out of $x^{3} \ln (2)$ .

$\frac{x^{2} \cdot 3}{x^{2} (x \ln (2))}$

Step 3.2.4.3.2.2

Cancel the common factor.

$\frac{x^{2} \cdot 3}{x^{2} (x \ln (2))}$

Step 3.2.4.3.2.3

Rewrite the expression.

$\frac{3}{x \ln (2)}$

Step 4

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

$y' = \frac{3}{x \ln (2)}$

Step 5

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

$\frac{d y}{d x} = \frac{3}{x \ln (2)}$