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Derivative of:
Derivative of atan(sqrt(x))
Derivative of 3*x^5
Derivative of 3*x^2*e^x
Derivative of 3*cos(3*x)
Identical expressions
sin(five *x)^(six)
sinus of (5 multiply by x) to the power of (6)
sinus of (five multiply by x) to the power of (six)
sin(5*x)(6)
sin5*x6
sin(5x)^(6)
sin(5x)(6)
sin5x6
sin5x^6

The derivative of the function/ sin(5*x)^(6)

Derivative of sin(5*x)^(6)

The solution

You have entered [src]

   6     
sin (5*x)

$$\sin^{6}{\left(5 x \right)}$$

d /   6     \
--\sin (5*x)/
dx

$$\frac{d}{d x} \sin^{6}{\left(5 x \right)}$$

Transform

Detail solution

Let $u = \sin{\left(5 x \right)}$ .
Apply the power rule: $u^{6}$ goes to $6 u^{5}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(5 x \right)}$ :
1. Let $u = 5 x$ .
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 $\frac{d}{d x} 5 x$ :
  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: $5$
  The result of the chain rule is:
  
  $5 \cos{\left(5 x \right)}$
The result of the chain rule is:

$30 \sin^{5}{\left(5 x \right)} \cos{\left(5 x \right)}$

The answer is:

$30 \sin^{5}{\left(5 x \right)} \cos{\left(5 x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      5              
30*sin (5*x)*cos(5*x)

$$30 \sin^{5}{\left(5 x \right)} \cos{\left(5 x \right)}$$

Simplify

The second derivative [src]

       4      /     2             2     \
150*sin (5*x)*\- sin (5*x) + 5*cos (5*x)/

$$150 \left(- \sin^{2}{\left(5 x \right)} + 5 \cos^{2}{\left(5 x \right)}\right) \sin^{4}{\left(5 x \right)}$$

Simplify

The third derivative [src]

        3      /       2             2     \         
3000*sin (5*x)*\- 4*sin (5*x) + 5*cos (5*x)/*cos(5*x)

$$3000 \left(- 4 \sin^{2}{\left(5 x \right)} + 5 \cos^{2}{\left(5 x \right)}\right) \sin^{3}{\left(5 x \right)} \cos{\left(5 x \right)}$$

Simplify

The graph

Derivative of sin(5*x)^(6)