Calculus Examples

$y = 5 \cos (x) \tan (x)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (5 \cos (x) \tan (x))$

Step 2

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

$y'$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

Since $5$ is constant with respect to $x$ , the derivative of $5 \cos (x) \tan (x)$ with respect to $x$ is $5 \frac{d}{d x} [\cos (x) \tan (x)]$ .

$5 \frac{d}{d x} [\cos (x) \tan (x)]$

Step 3.2

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 $f (x) = \cos (x)$ and $g (x) = \tan (x)$ .

$5 (\cos (x) \frac{d}{d x} [\tan (x)] + \tan (x) \frac{d}{d x} [\cos (x)])$

Step 3.3

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

$5 (\cos (x) \sec^{2} (x) + \tan (x) \frac{d}{d x} [\cos (x)])$

Step 3.4

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

$5 (\cos (x) \sec^{2} (x) + \tan (x) (- \sin (x)))$

Step 3.5

Simplify.

Tap for more steps...

Step 3.5.1

Apply the distributive property.

$5 (\cos (x) \sec^{2} (x)) + 5 (\tan (x) (- \sin (x)))$

Step 3.5.2

Multiply $- 1$ by $5$ .

$5 \cos (x) \sec^{2} (x) - 5 \tan (x) \sin (x)$

Step 3.5.3

Reorder terms.

$5 \sec^{2} (x) \cos (x) - 5 \sin (x) \tan (x)$

Step 3.5.4

Simplify each term.

Tap for more steps...

Step 3.5.4.1

Rewrite $\sec (x)$ in terms of sines and cosines.

$5 {(\frac{1}{\cos (x)})}^{2} \cos (x) - 5 \sin (x) \tan (x)$

Step 3.5.4.2

Apply the product rule to $\frac{1}{\cos (x)}$ .

$5 \frac{1^{2}}{\cos^{2} (x)} \cos (x) - 5 \sin (x) \tan (x)$

Step 3.5.4.3

One to any power is one.

$5 \frac{1}{\cos^{2} (x)} \cos (x) - 5 \sin (x) \tan (x)$

Step 3.5.4.4

Combine $5$ and $\frac{1}{\cos^{2} (x)}$ .

$\frac{5}{\cos^{2} (x)} \cos (x) - 5 \sin (x) \tan (x)$

Step 3.5.4.5

Cancel the common factor of $\cos (x)$ .

Tap for more steps...

Step 3.5.4.5.1

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

$\frac{5}{\cos (x) \cos (x)} \cos (x) - 5 \sin (x) \tan (x)$

Step 3.5.4.5.2

Cancel the common factor.

$\frac{5}{\cos (x) \cos (x)} \cos (x) - 5 \sin (x) \tan (x)$

Step 3.5.4.5.3

Rewrite the expression.

$\frac{5}{\cos (x)} - 5 \sin (x) \tan (x)$

Step 3.5.4.6

Rewrite $\tan (x)$ in terms of sines and cosines.

$\frac{5}{\cos (x)} - 5 \sin (x) \frac{\sin (x)}{\cos (x)}$

Step 3.5.4.7

Multiply $- 5 \sin (x) \frac{\sin (x)}{\cos (x)}$ .

Tap for more steps...

Step 3.5.4.7.1

Combine $\frac{\sin (x)}{\cos (x)}$ and $- 5$ .

$\frac{5}{\cos (x)} + \frac{\sin (x) \cdot - 5}{\cos (x)} \sin (x)$

Step 3.5.4.7.2

Combine $\frac{\sin (x) \cdot - 5}{\cos (x)}$ and $\sin (x)$ .

$\frac{5}{\cos (x)} + \frac{\sin (x) \cdot - 5 \sin (x)}{\cos (x)}$

Step 3.5.4.7.3

Raise $\sin (x)$ to the power of $1$ .

$\frac{5}{\cos (x)} + \frac{- 5 (\sin^{1} (x) \sin (x))}{\cos (x)}$

Step 3.5.4.7.4

Raise $\sin (x)$ to the power of $1$ .

$\frac{5}{\cos (x)} + \frac{- 5 (\sin^{1} (x) \sin^{1} (x))}{\cos (x)}$

Step 3.5.4.7.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{5}{\cos (x)} + \frac{- 5 {\sin (x)}^{1 + 1}}{\cos (x)}$

Step 3.5.4.7.6

Add $1$ and $1$ .

$\frac{5}{\cos (x)} + \frac{- 5 \sin^{2} (x)}{\cos (x)}$

Step 3.5.4.8

Move the negative in front of the fraction.

$\frac{5}{\cos (x)} - \frac{5 \sin^{2} (x)}{\cos (x)}$

Step 3.5.5

Combine the numerators over the common denominator.

$\frac{5 - 5 \sin^{2} (x)}{\cos (x)}$

Step 3.5.6

Factor $5$ out of $5$ .

$\frac{5 (1) - 5 \sin^{2} (x)}{\cos (x)}$

Step 3.5.7

Factor $5$ out of $- 5 \sin^{2} (x)$ .

$\frac{5 (1) + 5 (- \sin^{2} (x))}{\cos (x)}$

Step 3.5.8

Factor $5$ out of $5 (1) + 5 (- \sin^{2} (x))$ .

$\frac{5 (1 - \sin^{2} (x))}{\cos (x)}$

Step 3.5.9

Apply pythagorean identity.

$\frac{5 \cos^{2} (x)}{\cos (x)}$

Step 3.5.10

Cancel the common factor of $\cos^{2} (x)$ and $\cos (x)$ .

Tap for more steps...

Step 3.5.10.1

Factor $\cos (x)$ out of $5 \cos^{2} (x)$ .

$\frac{\cos (x) (5 \cos (x))}{\cos (x)}$

Step 3.5.10.2

Cancel the common factors.

Tap for more steps...

Step 3.5.10.2.1

Multiply by $1$ .

$\frac{\cos (x) (5 \cos (x))}{\cos (x) \cdot 1}$

Step 3.5.10.2.2

Cancel the common factor.

$\frac{\cos (x) (5 \cos (x))}{\cos (x) \cdot 1}$

Step 3.5.10.2.3

Rewrite the expression.

$\frac{5 \cos (x)}{1}$

Step 3.5.10.2.4

Divide $5 \cos (x)$ by $1$ .

$5 \cos (x)$

Step 4

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

$y' = 5 \cos (x)$

Step 5

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

$\frac{d y}{d x} = 5 \cos (x)$