Calculus Examples

$y = \frac{2 x^{2} + 3}{\cos (x)}$

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 $f (x) = 2 x^{2} + 3$ and $g (x) = \cos (x)$ .

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

Step 2

Differentiate.

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Multiply $2$ by $2$ .

$\frac{\cos (x) (4 x + \frac{d}{d x} [3]) - (2 x^{2} + 3) \frac{d}{d x} [\cos (x)]}{\cos^{2} (x)}$

Step 2.5

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

$\frac{\cos (x) (4 x + 0) - (2 x^{2} + 3) \frac{d}{d x} [\cos (x)]}{\cos^{2} (x)}$

Step 2.6

Add $4 x$ and $0$ .

$\frac{\cos (x) (4 x) - (2 x^{2} + 3) \frac{d}{d x} [\cos (x)]}{\cos^{2} (x)}$

Step 3

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

$\frac{\cos (x) (4 x) - (2 x^{2} + 3) (- \sin (x))}{\cos^{2} (x)}$

Step 4

Multiply.

Tap for more steps...

Step 4.1

Multiply $- 1$ by $- 1$ .

$\frac{\cos (x) (4 x) + 1 (2 x^{2} + 3) \sin (x)}{\cos^{2} (x)}$

Step 4.2

Multiply $2 x^{2} + 3$ by $1$ .

$\frac{\cos (x) (4 x) + (2 x^{2} + 3) \sin (x)}{\cos^{2} (x)}$

Step 5

Simplify.

Tap for more steps...

Step 5.1

Apply the distributive property.

$\frac{\cos (x) (4 x) + 2 x^{2} \sin (x) + 3 \sin (x)}{\cos^{2} (x)}$

Step 5.2

Simplify the numerator.

Tap for more steps...

Step 5.2.1

Rewrite using the commutative property of multiplication.

$\frac{4 \cos (x) x + 2 x^{2} \sin (x) + 3 \sin (x)}{\cos^{2} (x)}$

Step 5.2.2

Reorder factors in $4 \cos (x) x + 2 x^{2} \sin (x) + 3 \sin (x)$ .

$\frac{4 x \cos (x) + 2 x^{2} \sin (x) + 3 \sin (x)}{\cos^{2} (x)}$