Calculus Examples

$g (x) = (2 - x) \tan^{2} (x)$

Step 1

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

$(2 - x) \frac{d}{d x} [\tan^{2} (x)] + \tan^{2} (x) \frac{d}{d x} [2 - x]$

Step 2

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

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Step 2.1

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

$(2 - x) (\frac{d}{d u} [u^{2}] \frac{d}{d x} [\tan (x)]) + \tan^{2} (x) \frac{d}{d x} [2 - x]$

Step 2.2

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

$(2 - x) (2 u \frac{d}{d x} [\tan (x)]) + \tan^{2} (x) \frac{d}{d x} [2 - x]$

Step 2.3

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

$(2 - x) (2 \tan (x) \frac{d}{d x} [\tan (x)]) + \tan^{2} (x) \frac{d}{d x} [2 - x]$

Step 3

Move $2$ to the left of $2 - x$ .

$2 \cdot (2 - x) \tan (x) \frac{d}{d x} [\tan (x)] + \tan^{2} (x) \frac{d}{d x} [2 - x]$

Step 4

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

$2 (2 - x) \tan (x) \sec^{2} (x) + \tan^{2} (x) \frac{d}{d x} [2 - x]$

Step 5

Differentiate.

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Step 5.1

By the Sum Rule, the derivative of $2 - x$ with respect to $x$ is $\frac{d}{d x} [2] + \frac{d}{d x} [- x]$ .

$2 (2 - x) \tan (x) \sec^{2} (x) + \tan^{2} (x) (\frac{d}{d x} [2] + \frac{d}{d x} [- x])$

Step 5.2

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$2 (2 - x) \tan (x) \sec^{2} (x) + \tan^{2} (x) (0 + \frac{d}{d x} [- x])$

Step 5.3

Add $0$ and $\frac{d}{d x} [- x]$ .

$2 (2 - x) \tan (x) \sec^{2} (x) + \tan^{2} (x) \frac{d}{d x} [- x]$

Step 5.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$2 (2 - x) \tan (x) \sec^{2} (x) + \tan^{2} (x) (- \frac{d}{d x} [x])$

Step 5.5

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

$2 (2 - x) \tan (x) \sec^{2} (x) + \tan^{2} (x) (- 1 \cdot 1)$

Step 5.6

Simplify the expression.

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Step 5.6.1

Multiply $- 1$ by $1$ .

$2 (2 - x) \tan (x) \sec^{2} (x) + \tan^{2} (x) \cdot - 1$

Step 5.6.2

Move $- 1$ to the left of $\tan^{2} (x)$ .

$2 (2 - x) \tan (x) \sec^{2} (x) - 1 \cdot \tan^{2} (x)$

Step 5.6.3

Rewrite $- 1 \tan^{2} (x)$ as $- \tan^{2} (x)$ .

$2 (2 - x) \tan (x) \sec^{2} (x) - \tan^{2} (x)$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$(2 \cdot 2 + 2 (- x)) \tan (x) \sec^{2} (x) - \tan^{2} (x)$

Step 6.2

Apply the distributive property.

$(2 \cdot 2 \tan (x) + 2 (- x) \tan (x)) \sec^{2} (x) - \tan^{2} (x)$

Step 6.3

Apply the distributive property.

$2 \cdot 2 \tan (x) \sec^{2} (x) + 2 (- x) \tan (x) \sec^{2} (x) - \tan^{2} (x)$

Step 6.4

Combine terms.

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Step 6.4.1

Multiply $2$ by $2$ .

$4 \tan (x) \sec^{2} (x) + 2 (- x) \tan (x) \sec^{2} (x) - \tan^{2} (x)$

Step 6.4.2

Multiply $- 1$ by $2$ .

$4 \tan (x) \sec^{2} (x) - 2 x \tan (x) \sec^{2} (x) - \tan^{2} (x)$

Step 6.5

Reorder terms.

$4 \sec^{2} (x) \tan (x) - 2 x \sec^{2} (x) \tan (x) - \tan^{2} (x)$