Calculus Examples

$r = \tan^{2} (3 - x^{2})$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (r) = \frac{d}{d x} (\tan^{2} (3 - x^{2}))$

Step 2

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

$r'$

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 $f (x) = x^{2}$ and $g (x) = \tan (3 - x^{2})$ .

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Step 3.1.1

To apply the Chain Rule, set $u_{1}$ as $\tan (3 - x^{2})$ .

$\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [\tan (3 - x^{2})]$

Step 3.1.2

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

$2 u_{1} \frac{d}{d x} [\tan (3 - x^{2})]$

Step 3.1.3

Replace all occurrences of $u_{1}$ with $\tan (3 - x^{2})$ .

$2 \tan (3 - x^{2}) \frac{d}{d x} [\tan (3 - x^{2})]$

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 $f (x) = \tan (x)$ and $g (x) = 3 - x^{2}$ .

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Step 3.2.1

To apply the Chain Rule, set $u_{2}$ as $3 - x^{2}$ .

$2 \tan (3 - x^{2}) (\frac{d}{d u_{2}} [\tan (u_{2})] \frac{d}{d x} [3 - x^{2}])$

Step 3.2.2

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

$2 \tan (3 - x^{2}) (\sec^{2} (u_{2}) \frac{d}{d x} [3 - x^{2}])$

Step 3.2.3

Replace all occurrences of $u_{2}$ with $3 - x^{2}$ .

$2 \tan (3 - x^{2}) (\sec^{2} (3 - x^{2}) \frac{d}{d x} [3 - x^{2}])$

Step 3.3

Differentiate.

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Step 3.3.1

By the Sum Rule, the derivative of $3 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [3] + \frac{d}{d x} [- x^{2}]$ .

$2 \tan (3 - x^{2}) \sec^{2} (3 - x^{2}) (\frac{d}{d x} [3] + \frac{d}{d x} [- x^{2}])$

Step 3.3.2

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$2 \tan (3 - x^{2}) \sec^{2} (3 - x^{2}) (0 + \frac{d}{d x} [- x^{2}])$

Step 3.3.3

Add $0$ and $\frac{d}{d x} [- x^{2}]$ .

$2 \tan (3 - x^{2}) \sec^{2} (3 - x^{2}) \frac{d}{d x} [- x^{2}]$

Step 3.3.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$2 \tan (3 - x^{2}) \sec^{2} (3 - x^{2}) (- \frac{d}{d x} [x^{2}])$

Step 3.3.5

Multiply $- 1$ by $2$ .

$- 2 \tan (3 - x^{2}) \sec^{2} (3 - x^{2}) \frac{d}{d x} [x^{2}]$

Step 3.3.6

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

$- 2 \tan (3 - x^{2}) \sec^{2} (3 - x^{2}) (2 x)$

Step 3.3.7

Simplify the expression.

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Step 3.3.7.1

Multiply $2$ by $- 2$ .

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

Step 3.3.7.2

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

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

Step 4

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

$r' = - 4 x \sec^{2} (3 - x^{2}) \tan (3 - x^{2})$

Step 5

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

$\frac{d r}{d x} = - 4 x \sec^{2} (3 - x^{2}) \tan (3 - x^{2})$