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Derivative of 1/z
Derivative of f(x)=2x
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Derivative of x^3*cot(4*x)
Identical expressions
tan(three *x-pi/ six)
tangent of (3 multiply by x minus Pi divide by 6)
tangent of (three multiply by x minus Pi divide by six)
tan(3x-pi/6)
tan3x-pi/6
tan(3*x-pi divide by 6)
Similar expressions
tan(3*x+pi/6)
What you mean?
tan((3*x - pi)/6)
tan(3*(x - pi)/6)
tan(3*x - pi/6)
tan(3*x - pi/6)

The derivative of the function/ tan(3*x-pi/6)

You entered:

tan(3*x-pi/6)

What you mean?

tan((3*x - pi)/6)

tan(3*(x - pi)/6)

tan(3*x - pi/6)

tan(3*x - pi/6)

Derivative of tan(3*x-pi/6)

The solution

You have entered [src]

   /      pi\
tan|3*x - --|
   \      6 /

$$\tan{\left(3 x - \frac{\pi}{6} \right)}$$

Transform

Detail solution

Rewrite the function to be differentiated:

$\tan{\left(3 x - \frac{\pi}{6} \right)} = \frac{\sin{\left(3 x - \frac{\pi}{6} \right)}}{\cos{\left(3 x - \frac{\pi}{6} \right)}}$
Apply the quotient rule, which is:

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

$f{\left(x \right)} = \sin{\left(3 x - \frac{\pi}{6} \right)}$ and $g{\left(x \right)} = \cos{\left(3 x - \frac{\pi}{6} \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = 3 x - \frac{\pi}{6}$ .
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 $\frac{d}{d x} \left(3 x - \frac{\pi}{6}\right)$ :
  1. Differentiate $3 x - \frac{\pi}{6}$ 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: $3$
    2. The derivative of the constant $- \frac{\pi}{6}$ is zero.
    The result is: $3$
  The result of the chain rule is:
  
  $3 \cos{\left(3 x - \frac{\pi}{6} \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 3 x - \frac{\pi}{6}$ .
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 $\frac{d}{d x} \left(3 x - \frac{\pi}{6}\right)$ :
  1. Differentiate $3 x - \frac{\pi}{6}$ 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: $3$
    2. The derivative of the constant $- \frac{\pi}{6}$ is zero.
    The result is: $3$
  The result of the chain rule is:
  
  $- 3 \sin{\left(3 x - \frac{\pi}{6} \right)}$
Now plug in to the quotient rule:

$\frac{3 \sin^{2}{\left(3 x - \frac{\pi}{6} \right)} + 3 \cos^{2}{\left(3 x - \frac{\pi}{6} \right)}}{\cos^{2}{\left(3 x - \frac{\pi}{6} \right)}}$
Now simplify:

$\frac{3}{\sin^{2}{\left(3 x + \frac{\pi}{3} \right)}}$

The answer is:

$\frac{3}{\sin^{2}{\left(3 x + \frac{\pi}{3} \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

         2/      pi\
3 + 3*tan |3*x - --|
          \      6 /

$$3 \tan^{2}{\left(3 x - \frac{\pi}{6} \right)} + 3$$

Simplify

The second derivative [src]

    /       2/      pi\\    /      pi\
-18*|1 + cot |3*x + --||*cot|3*x + --|
    \        \      3 //    \      3 /

$$- 18 \left(\cot^{2}{\left(3 x + \frac{\pi}{3} \right)} + 1\right) \cot{\left(3 x + \frac{\pi}{3} \right)}$$

Simplify

The third derivative [src]

   /       2/      pi\\ /         2/      pi\\
54*|1 + cot |3*x + --||*|1 + 3*cot |3*x + --||
   \        \      3 // \          \      3 //

$$54 \left(\cot^{2}{\left(3 x + \frac{\pi}{3} \right)} + 1\right) \left(3 \cot^{2}{\left(3 x + \frac{\pi}{3} \right)} + 1\right)$$

Simplify

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What you mean?

Derivative of tan(3*x-pi/6)

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The solution