Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of 1/z
Derivative of f(x)=2x
Derivative of x^3*(x^2-1)^2
Derivative of x^3*cot(4*x)
Identical expressions
tan(five *x+pi/ eight)
tangent of (5 multiply by x plus Pi divide by 8)
tangent of (five multiply by x plus Pi divide by eight)
tan(5x+pi/8)
tan5x+pi/8
tan(5*x+pi divide by 8)
Similar expressions
tan(5*x-pi/8)

The derivative of the function/ tan(5*x+pi/8)

Derivative of tan(5*x+pi/8)

The solution

You have entered [src]

   /      pi\
tan|5*x + --|
   \      8 /

$$\tan{\left(5 x + \frac{\pi}{8} \right)}$$

tan(5*x + pi/8)

Detail solution

Rewrite the function to be differentiated:

$\tan{\left(5 x + \frac{\pi}{8} \right)} = \frac{\sin{\left(5 x + \frac{\pi}{8} \right)}}{\cos{\left(5 x + \frac{\pi}{8} \right)}}$
Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \sin{\left(5 x + \frac{\pi}{8} \right)}$ and $g{\left(x \right)} = \cos{\left(5 x + \frac{\pi}{8} \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = 5 x + \frac{\pi}{8}$ .
2. The derivative of sine is cosine:
  
  $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(5 x + \frac{\pi}{8}\right)$ :
  1. Differentiate $5 x + \frac{\pi}{8}$ term by term:
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $5$
    2. The derivative of the constant $\frac{\pi}{8}$ is zero.
    The result is: $5$
  The result of the chain rule is:
  
  $5 \cos{\left(5 x + \frac{\pi}{8} \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 5 x + \frac{\pi}{8}$ .
2. The derivative of cosine is negative sine:
  
  $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(5 x + \frac{\pi}{8}\right)$ :
  1. Differentiate $5 x + \frac{\pi}{8}$ term by term:
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $5$
    2. The derivative of the constant $\frac{\pi}{8}$ is zero.
    The result is: $5$
  The result of the chain rule is:
  
  $- 5 \sin{\left(5 x + \frac{\pi}{8} \right)}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

         2/      pi\
5 + 5*tan |5*x + --|
          \      8 /

$$5 \tan^{2}{\left(5 x + \frac{\pi}{8} \right)} + 5$$

Simplify

The second derivative [src]

   /       2/      pi\\    /      pi\
50*|1 + tan |5*x + --||*tan|5*x + --|
   \        \      8 //    \      8 /

$$50 \left(\tan^{2}{\left(5 x + \frac{\pi}{8} \right)} + 1\right) \tan{\left(5 x + \frac{\pi}{8} \right)}$$

Simplify

The third derivative [src]

    /       2/      pi\\ /         2/      pi\\
250*|1 + tan |5*x + --||*|1 + 3*tan |5*x + --||
    \        \      8 // \          \      8 //

$$250 \left(\tan^{2}{\left(5 x + \frac{\pi}{8} \right)} + 1\right) \left(3 \tan^{2}{\left(5 x + \frac{\pi}{8} \right)} + 1\right)$$

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of tan(5*x+pi/8)

The graph:

Piecewise:

The solution