Derivative of 8secx-5cosx

You have entered [src]

8*sec(x) - 5*cos(x)

$$- 5 \cos{\left(x \right)} + 8 \sec{\left(x \right)}$$

d                      
--(8*sec(x) - 5*cos(x))
dx

$$\frac{d}{d x} \left(- 5 \cos{\left(x \right)} + 8 \sec{\left(x \right)}\right)$$

Transform

Detail solution

Differentiate $- 5 \cos{\left(x \right)} + 8 \sec{\left(x \right)}$ term by term:
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Rewrite the function to be differentiated:
    
    $\sec{\left(x \right)} = \frac{1}{\cos{\left(x \right)}}$
  2. Let $u = \cos{\left(x \right)}$ .
  3. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
  4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(x \right)}$ :
    1. The derivative of cosine is negative sine:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    The result of the chain rule is:
    
    $\frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  So, the result is: $\frac{8 \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. The derivative of cosine is negative sine:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    So, the result is: $- 5 \sin{\left(x \right)}$
  So, the result is: $5 \sin{\left(x \right)}$
The result is: $5 \sin{\left(x \right)} + \frac{8 \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
Now simplify:

$\left(5 + \frac{8}{\cos^{2}{\left(x \right)}}\right) \sin{\left(x \right)}$

The answer is:

$\left(5 + \frac{8}{\cos^{2}{\left(x \right)}}\right) \sin{\left(x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

5*sin(x) + 8*sec(x)*tan(x)

$$5 \sin{\left(x \right)} + 8 \tan{\left(x \right)} \sec{\left(x \right)}$$

Simplify

The second derivative [src]

                2               /       2   \       
5*cos(x) + 8*tan (x)*sec(x) + 8*\1 + tan (x)/*sec(x)

$$8 \left(\tan^{2}{\left(x \right)} + 1\right) \sec{\left(x \right)} + 5 \cos{\left(x \right)} + 8 \tan^{2}{\left(x \right)} \sec{\left(x \right)}$$

Simplify

The third derivative [src]

                 3                /       2   \              
-5*sin(x) + 8*tan (x)*sec(x) + 40*\1 + tan (x)/*sec(x)*tan(x)

$$40 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} \sec{\left(x \right)} - 5 \sin{\left(x \right)} + 8 \tan^{3}{\left(x \right)} \sec{\left(x \right)}$$

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of 8secx-5cosx

The graph:

Piecewise:

The solution