Calculus Examples

$f (x) = \frac{x^{2} + 1}{\cos (x)}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x^{2} + 1$ and $g (x) = \cos (x)$ .

$\frac{\cos (x) \frac{d}{d x} [x^{2} + 1] - (x^{2} + 1) \frac{d}{d x} [\cos (x)]}{\cos^{2} (x)}$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$ .

$\frac{\cos (x) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]) - (x^{2} + 1) \frac{d}{d x} [\cos (x)]}{\cos^{2} (x)}$

Step 2.2

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

$\frac{\cos (x) (2 x + \frac{d}{d x} [1]) - (x^{2} + 1) \frac{d}{d x} [\cos (x)]}{\cos^{2} (x)}$

Step 2.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{\cos (x) (2 x + 0) - (x^{2} + 1) \frac{d}{d x} [\cos (x)]}{\cos^{2} (x)}$

Step 2.4

Add $2 x$ and $0$ .

$\frac{\cos (x) (2 x) - (x^{2} + 1) \frac{d}{d x} [\cos (x)]}{\cos^{2} (x)}$

Step 3

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$\frac{\cos (x) (2 x) - (x^{2} + 1) (- \sin (x))}{\cos^{2} (x)}$

Step 4

Multiply.

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Step 4.1

Multiply $- 1$ by $- 1$ .

$\frac{\cos (x) (2 x) + 1 (x^{2} + 1) \sin (x)}{\cos^{2} (x)}$

Step 4.2

Multiply $x^{2} + 1$ by $1$ .

$\frac{\cos (x) (2 x) + (x^{2} + 1) \sin (x)}{\cos^{2} (x)}$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$\frac{\cos (x) (2 x) + x^{2} \sin (x) + 1 \sin (x)}{\cos^{2} (x)}$

Step 5.2

Simplify the numerator.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Rewrite using the commutative property of multiplication.

$\frac{2 \cos (x) x + x^{2} \sin (x) + 1 \sin (x)}{\cos^{2} (x)}$

Step 5.2.1.2

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

$\frac{2 \cos (x) x + x^{2} \sin (x) + \sin (x)}{\cos^{2} (x)}$

Step 5.2.2

Reorder factors in $2 \cos (x) x + x^{2} \sin (x) + \sin (x)$ .

$\frac{2 x \cos (x) + x^{2} \sin (x) + \sin (x)}{\cos^{2} (x)}$