Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of (4-3*x)^7
Derivative of 3x²-2
Derivative of (x-5)^2*e^x-7
Derivative of x/(1+e^x)
Identical expressions
tg^ five *(x+ two)
tg to the power of 5 multiply by (x plus 2)
tg to the power of five multiply by (x plus two)
tg5*(x+2)
tg5*x+2
tg⁵*(x+2)
tg^5(x+2)
tg5(x+2)
tg5x+2
tg^5x+2
Similar expressions
tg^5*(x-2)

The derivative of the function/ tg^5*(x+2)

Derivative of tg^5*(x+2)

The solution

You have entered [src]

   5       
tan (x + 2)

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

tan(x + 2)^5

Detail solution

Let $u = \tan{\left(x + 2 \right)}$ .
Apply the power rule: $u^{5}$ goes to $5 u^{4}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(x + 2 \right)}$ :
1. Rewrite the function to be differentiated:
  
  $\tan{\left(x + 2 \right)} = \frac{\sin{\left(x + 2 \right)}}{\cos{\left(x + 2 \right)}}$
2. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sin{\left(x + 2 \right)}$ and $g{\left(x \right)} = \cos{\left(x + 2 \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = x + 2$ .
  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 $\frac{d}{d x} \left(x + 2\right)$ :
    1. Differentiate $x + 2$ term by term:
      1. Apply the power rule: $x$ goes to $1$
      2. The derivative of the constant $2$ is zero.
      The result is: $1$
    The result of the chain rule is:
    
    $\cos{\left(x + 2 \right)}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = x + 2$ .
  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 $\frac{d}{d x} \left(x + 2\right)$ :
    1. Differentiate $x + 2$ term by term:
      1. Apply the power rule: $x$ goes to $1$
      2. The derivative of the constant $2$ is zero.
      The result is: $1$
    The result of the chain rule is:
    
    $- \sin{\left(x + 2 \right)}$
  Now plug in to the quotient rule:
  
  $\frac{\sin^{2}{\left(x + 2 \right)} + \cos^{2}{\left(x + 2 \right)}}{\cos^{2}{\left(x + 2 \right)}}$
The result of the chain rule is:

$\frac{5 \left(\sin^{2}{\left(x + 2 \right)} + \cos^{2}{\left(x + 2 \right)}\right) \tan^{4}{\left(x + 2 \right)}}{\cos^{2}{\left(x + 2 \right)}}$
Now simplify:

$\frac{5 \tan^{4}{\left(x + 2 \right)}}{\cos^{2}{\left(x + 2 \right)}}$

The answer is:

$\frac{5 \tan^{4}{\left(x + 2 \right)}}{\cos^{2}{\left(x + 2 \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   4        /         2       \
tan (x + 2)*\5 + 5*tan (x + 2)/

$$\left(5 \tan^{2}{\left(x + 2 \right)} + 5\right) \tan^{4}{\left(x + 2 \right)}$$

Simplify

The second derivative [src]

      3        /       2       \ /         2       \
10*tan (2 + x)*\1 + tan (2 + x)/*\2 + 3*tan (2 + x)/

$$10 \left(\tan^{2}{\left(x + 2 \right)} + 1\right) \left(3 \tan^{2}{\left(x + 2 \right)} + 2\right) \tan^{3}{\left(x + 2 \right)}$$

Simplify

The third derivative [src]

                                 /                                   2                                   \
      2        /       2       \ |     4            /       2       \          2        /       2       \|
10*tan (2 + x)*\1 + tan (2 + x)/*\2*tan (2 + x) + 6*\1 + tan (2 + x)/  + 13*tan (2 + x)*\1 + tan (2 + x)//

$$10 \left(\tan^{2}{\left(x + 2 \right)} + 1\right) \left(6 \left(\tan^{2}{\left(x + 2 \right)} + 1\right)^{2} + 13 \left(\tan^{2}{\left(x + 2 \right)} + 1\right) \tan^{2}{\left(x + 2 \right)} + 2 \tan^{4}{\left(x + 2 \right)}\right) \tan^{2}{\left(x + 2 \right)}$$

Simplify

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of tg^5*(x+2)

The graph:

Piecewise:

The solution