Calculus Examples

$\tan (x) (4 \sin (x) + 5 \cos (x))$

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 $f (x) = \tan (x)$ and $g (x) = 4 \sin (x) + 5 \cos (x)$ .

$\tan (x) \frac{d}{d x} [4 \sin (x) + 5 \cos (x)] + (4 \sin (x) + 5 \cos (x)) \frac{d}{d x} [\tan (x)]$

Step 2

Differentiate.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $4 \sin (x) + 5 \cos (x)$ with respect to $x$ is $\frac{d}{d x} [4 \sin (x)] + \frac{d}{d x} [5 \cos (x)]$ .

$\tan (x) (\frac{d}{d x} [4 \sin (x)] + \frac{d}{d x} [5 \cos (x)]) + (4 \sin (x) + 5 \cos (x)) \frac{d}{d x} [\tan (x)]$

Step 2.2

Since $4$ is constant with respect to $x$ , the derivative of $4 \sin (x)$ with respect to $x$ is $4 \frac{d}{d x} [\sin (x)]$ .

$\tan (x) (4 \frac{d}{d x} [\sin (x)] + \frac{d}{d x} [5 \cos (x)]) + (4 \sin (x) + 5 \cos (x)) \frac{d}{d x} [\tan (x)]$

Step 3

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

$\tan (x) (4 \cos (x) + \frac{d}{d x} [5 \cos (x)]) + (4 \sin (x) + 5 \cos (x)) \frac{d}{d x} [\tan (x)]$

Step 4

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

$\tan (x) (4 \cos (x) + 5 \frac{d}{d x} [\cos (x)]) + (4 \sin (x) + 5 \cos (x)) \frac{d}{d x} [\tan (x)]$

Step 5

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

$\tan (x) (4 \cos (x) + 5 (- \sin (x))) + (4 \sin (x) + 5 \cos (x)) \frac{d}{d x} [\tan (x)]$

Step 6

Multiply $- 1$ by $5$ .

$\tan (x) (4 \cos (x) - 5 \sin (x)) + (4 \sin (x) + 5 \cos (x)) \frac{d}{d x} [\tan (x)]$

Step 7

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

$\tan (x) (4 \cos (x) - 5 \sin (x)) + (4 \sin (x) + 5 \cos (x)) \sec^{2} (x)$

Step 8

Simplify.

Tap for more steps...

Step 8.1

Apply the distributive property.

$\tan (x) (4 \cos (x)) + \tan (x) (- 5 \sin (x)) + (4 \sin (x) + 5 \cos (x)) \sec^{2} (x)$

Step 8.2

Apply the distributive property.

$\tan (x) (4 \cos (x)) + \tan (x) (- 5 \sin (x)) + 4 \sin (x) \sec^{2} (x) + 5 \cos (x) \sec^{2} (x)$

Step 8.3

Move $4$ to the left of $\tan (x)$ .

$4 \tan (x) \cos (x) + \tan (x) (- 5 \sin (x)) + 4 \sin (x) \sec^{2} (x) + 5 \cos (x) \sec^{2} (x)$

Step 8.4

Reorder terms.

$4 \cos (x) \tan (x) - 5 \sin (x) \tan (x) + 4 \sec^{2} (x) \sin (x) + 5 \sec^{2} (x) \cos (x)$

Step 8.5

Simplify each term.

Tap for more steps...

Step 8.5.1

Rewrite in terms of sines and cosines, then cancel the common factors.

Tap for more steps...

Step 8.5.1.1

Add parentheses.

$4 (\cos (x) \tan (x)) - 5 \sin (x) \tan (x) + 4 \sec^{2} (x) \sin (x) + 5 \sec^{2} (x) \cos (x)$

Step 8.5.1.2

Reorder $\cos (x)$ and $\tan (x)$ .

$4 (\tan (x) \cos (x)) - 5 \sin (x) \tan (x) + 4 \sec^{2} (x) \sin (x) + 5 \sec^{2} (x) \cos (x)$

Step 8.5.1.3

Rewrite $4 \cos (x) \tan (x)$ in terms of sines and cosines.

$4 (\frac{\sin (x)}{\cos (x)} \cos (x)) - 5 \sin (x) \tan (x) + 4 \sec^{2} (x) \sin (x) + 5 \sec^{2} (x) \cos (x)$

Step 8.5.1.4

Cancel the common factors.

$4 \sin (x) - 5 \sin (x) \tan (x) + 4 \sec^{2} (x) \sin (x) + 5 \sec^{2} (x) \cos (x)$

Step 8.5.2

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

$4 \sin (x) - 5 \sin (x) \frac{\sin (x)}{\cos (x)} + 4 \sec^{2} (x) \sin (x) + 5 \sec^{2} (x) \cos (x)$

Step 8.5.3

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

Tap for more steps...

Step 8.5.3.1

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

$4 \sin (x) + \frac{\sin (x) \cdot - 5}{\cos (x)} \sin (x) + 4 \sec^{2} (x) \sin (x) + 5 \sec^{2} (x) \cos (x)$

Step 8.5.3.2

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

$4 \sin (x) + \frac{\sin (x) \cdot - 5 \sin (x)}{\cos (x)} + 4 \sec^{2} (x) \sin (x) + 5 \sec^{2} (x) \cos (x)$

Step 8.5.3.3

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

$4 \sin (x) + \frac{- 5 (\sin^{1} (x) \sin (x))}{\cos (x)} + 4 \sec^{2} (x) \sin (x) + 5 \sec^{2} (x) \cos (x)$

Step 8.5.3.4

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

$4 \sin (x) + \frac{- 5 (\sin^{1} (x) \sin^{1} (x))}{\cos (x)} + 4 \sec^{2} (x) \sin (x) + 5 \sec^{2} (x) \cos (x)$

Step 8.5.3.5

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

$4 \sin (x) + \frac{- 5 {\sin (x)}^{1 + 1}}{\cos (x)} + 4 \sec^{2} (x) \sin (x) + 5 \sec^{2} (x) \cos (x)$

Step 8.5.3.6

Add $1$ and $1$ .

$4 \sin (x) + \frac{- 5 \sin^{2} (x)}{\cos (x)} + 4 \sec^{2} (x) \sin (x) + 5 \sec^{2} (x) \cos (x)$

Step 8.5.4

Move the negative in front of the fraction.

$4 \sin (x) - \frac{5 \sin^{2} (x)}{\cos (x)} + 4 \sec^{2} (x) \sin (x) + 5 \sec^{2} (x) \cos (x)$

Step 8.5.5

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

$4 \sin (x) - \frac{5 \sin^{2} (x)}{\cos (x)} + 4 {(\frac{1}{\cos (x)})}^{2} \sin (x) + 5 \sec^{2} (x) \cos (x)$

Step 8.5.6

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

$4 \sin (x) - \frac{5 \sin^{2} (x)}{\cos (x)} + 4 \frac{1^{2}}{\cos^{2} (x)} \sin (x) + 5 \sec^{2} (x) \cos (x)$

Step 8.5.7

One to any power is one.

$4 \sin (x) - \frac{5 \sin^{2} (x)}{\cos (x)} + 4 \frac{1}{\cos^{2} (x)} \sin (x) + 5 \sec^{2} (x) \cos (x)$

Step 8.5.8

Combine $4$ and $\frac{1}{\cos^{2} (x)}$ .

$4 \sin (x) - \frac{5 \sin^{2} (x)}{\cos (x)} + \frac{4}{\cos^{2} (x)} \sin (x) + 5 \sec^{2} (x) \cos (x)$

Step 8.5.9

Combine $\frac{4}{\cos^{2} (x)}$ and $\sin (x)$ .

$4 \sin (x) - \frac{5 \sin^{2} (x)}{\cos (x)} + \frac{4 \sin (x)}{\cos^{2} (x)} + 5 \sec^{2} (x) \cos (x)$

Step 8.5.10

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

$4 \sin (x) - \frac{5 \sin^{2} (x)}{\cos (x)} + \frac{4 \sin (x)}{\cos^{2} (x)} + 5 {(\frac{1}{\cos (x)})}^{2} \cos (x)$

Step 8.5.11

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

$4 \sin (x) - \frac{5 \sin^{2} (x)}{\cos (x)} + \frac{4 \sin (x)}{\cos^{2} (x)} + 5 \frac{1^{2}}{\cos^{2} (x)} \cos (x)$

Step 8.5.12

One to any power is one.

$4 \sin (x) - \frac{5 \sin^{2} (x)}{\cos (x)} + \frac{4 \sin (x)}{\cos^{2} (x)} + 5 \frac{1}{\cos^{2} (x)} \cos (x)$

Step 8.5.13

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

$4 \sin (x) - \frac{5 \sin^{2} (x)}{\cos (x)} + \frac{4 \sin (x)}{\cos^{2} (x)} + \frac{5}{\cos^{2} (x)} \cos (x)$

Step 8.5.14

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

Tap for more steps...

Step 8.5.14.1

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

$4 \sin (x) - \frac{5 \sin^{2} (x)}{\cos (x)} + \frac{4 \sin (x)}{\cos^{2} (x)} + \frac{5}{\cos (x) \cos (x)} \cos (x)$

Step 8.5.14.2

Cancel the common factor.

$4 \sin (x) - \frac{5 \sin^{2} (x)}{\cos (x)} + \frac{4 \sin (x)}{\cos^{2} (x)} + \frac{5}{\cos (x) \cos (x)} \cos (x)$

Step 8.5.14.3

Rewrite the expression.

$4 \sin (x) - \frac{5 \sin^{2} (x)}{\cos (x)} + \frac{4 \sin (x)}{\cos^{2} (x)} + \frac{5}{\cos (x)}$

Step 8.6

Combine the numerators over the common denominator.

$4 \sin (x) + \frac{- 5 \sin^{2} (x) + 5}{\cos (x)} + \frac{4 \sin (x)}{\cos^{2} (x)}$

Step 8.7

Reorder $- 5 \sin^{2} (x)$ and $5$ .

$4 \sin (x) + \frac{5 - 5 \sin^{2} (x)}{\cos (x)} + \frac{4 \sin (x)}{\cos^{2} (x)}$

Step 8.8

Factor $5$ out of $5$ .

$4 \sin (x) + \frac{5 (1) - 5 \sin^{2} (x)}{\cos (x)} + \frac{4 \sin (x)}{\cos^{2} (x)}$

Step 8.9

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

$4 \sin (x) + \frac{5 (1) + 5 (- \sin^{2} (x))}{\cos (x)} + \frac{4 \sin (x)}{\cos^{2} (x)}$

Step 8.10

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

$4 \sin (x) + \frac{5 (1 - \sin^{2} (x))}{\cos (x)} + \frac{4 \sin (x)}{\cos^{2} (x)}$

Step 8.11

Apply pythagorean identity.

$4 \sin (x) + \frac{5 \cos^{2} (x)}{\cos (x)} + \frac{4 \sin (x)}{\cos^{2} (x)}$

Step 8.12

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

Tap for more steps...

Step 8.12.1

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

$4 \sin (x) + \frac{\cos (x) (5 \cos (x))}{\cos (x)} + \frac{4 \sin (x)}{\cos^{2} (x)}$

Step 8.12.2

Cancel the common factors.

Tap for more steps...

Step 8.12.2.1

Multiply by $1$ .

$4 \sin (x) + \frac{\cos (x) (5 \cos (x))}{\cos (x) \cdot 1} + \frac{4 \sin (x)}{\cos^{2} (x)}$

Step 8.12.2.2

Cancel the common factor.

$4 \sin (x) + \frac{\cos (x) (5 \cos (x))}{\cos (x) \cdot 1} + \frac{4 \sin (x)}{\cos^{2} (x)}$

Step 8.12.2.3

Rewrite the expression.

$4 \sin (x) + \frac{5 \cos (x)}{1} + \frac{4 \sin (x)}{\cos^{2} (x)}$

Step 8.12.2.4

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

$4 \sin (x) + 5 \cos (x) + \frac{4 \sin (x)}{\cos^{2} (x)}$

Step 8.13

Simplify each term.

Tap for more steps...

Step 8.13.1

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

$4 \sin (x) + 5 \cos (x) + \frac{4 \sin (x)}{\cos (x) \cos (x)}$

Step 8.13.2

Separate fractions.

$4 \sin (x) + 5 \cos (x) + \frac{4}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$

Step 8.13.3

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$4 \sin (x) + 5 \cos (x) + \frac{4}{\cos (x)} \tan (x)$

Step 8.13.4

Separate fractions.

$4 \sin (x) + 5 \cos (x) + \frac{4}{1} \cdot \frac{1}{\cos (x)} \tan (x)$

Step 8.13.5

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$4 \sin (x) + 5 \cos (x) + \frac{4}{1} \sec (x) \tan (x)$

Step 8.13.6

Divide $4$ by $1$ .

$4 \sin (x) + 5 \cos (x) + 4 \sec (x) \tan (x)$