Calculus Examples

$f (x) = \frac{1}{2} \cdot \tan (x^{4})$

Step 1

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{1}{2} \cdot \tan (x^{4})$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [\tan (x^{4})]$ .

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

Step 2

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

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Step 2.1

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

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

Step 2.2

The derivative of $\tan (u)$ with respect to $u$ is $\sec^{2} (u)$ .

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

Step 2.3

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

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

Step 3

Differentiate using the Power Rule.

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Step 3.1

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

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

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$\frac{\sec^{2} (x^{4})}{2} (4 x^{3})$

Step 3.3

Simplify terms.

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Step 3.3.1

Combine $4$ and $\frac{\sec^{2} (x^{4})}{2}$ .

$\frac{4 \sec^{2} (x^{4})}{2} x^{3}$

Step 3.3.2

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

$\frac{4 \sec^{2} (x^{4}) x^{3}}{2}$

Step 3.3.3

Cancel the common factor of $4$ and $2$ .

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Step 3.3.3.1

Factor $2$ out of $4 \sec^{2} (x^{4}) x^{3}$ .

$\frac{2 (2 \sec^{2} (x^{4}) x^{3})}{2}$

Step 3.3.3.2

Cancel the common factors.

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Step 3.3.3.2.1

Factor $2$ out of $2$ .

$\frac{2 (2 \sec^{2} (x^{4}) x^{3})}{2 (1)}$

Step 3.3.3.2.2

Cancel the common factor.

$\frac{2 (2 \sec^{2} (x^{4}) x^{3})}{2 \cdot 1}$

Step 3.3.3.2.3

Rewrite the expression.

$\frac{2 \sec^{2} (x^{4}) x^{3}}{1}$

Step 3.3.3.2.4

Divide $2 \sec^{2} (x^{4}) x^{3}$ by $1$ .

$2 \sec^{2} (x^{4}) x^{3}$

Step 3.3.4

Reorder the factors of $2 \sec^{2} (x^{4}) x^{3}$ .

$2 x^{3} \sec^{2} (x^{4})$