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Derivative of:
Derivative of x*2^x
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Derivative of 3*sin(2*x)
Identical expressions
sinx*cos^3x^ two
sinus of x multiply by co sinus of e of cubed x squared
sinus of x multiply by co sinus of e of cubed x to the power of two
sinx*cos3x2
sinx*cos³x²
sinx*cos to the power of 3x to the power of 2
sinxcos^3x^2
sinxcos3x2

The derivative of the function/ sinx*cos^3x^2

Derivative of sinx*cos^3x^2

The solution

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          9   
sin(x)*cos (x)

$$\sin{\left(x \right)} \cos^{9}{\left(x \right)}$$

Transform

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \sin{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of sine is cosine:
  
  $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
$g{\left(x \right)} = \cos^{9}{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \cos{\left(x \right)}$ .
2. Apply the power rule: $u^{9}$ goes to $9 u^{8}$
3. 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:
  
  $- 9 \sin{\left(x \right)} \cos^{8}{\left(x \right)}$
The result is: $- 9 \sin^{2}{\left(x \right)} \cos^{8}{\left(x \right)} + \cos^{10}{\left(x \right)}$
Now simplify:

$\left(5 \cos{\left(2 x \right)} - 4\right) \cos^{8}{\left(x \right)}$

The answer is:

$\left(5 \cos{\left(2 x \right)} - 4\right) \cos^{8}{\left(x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   10           8       2   
cos  (x) - 9*cos (x)*sin (x)

$$- 9 \sin^{2}{\left(x \right)} \cos^{8}{\left(x \right)} + \cos^{10}{\left(x \right)}$$

Simplify

The second derivative [src]

   7    /        2            2   \       
cos (x)*\- 28*cos (x) + 72*sin (x)/*sin(x)

$$\left(72 \sin^{2}{\left(x \right)} - 28 \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} \cos^{7}{\left(x \right)}$$

Simplify

The third derivative [src]

   6    /     4           2    /        2            2   \         2       2            2    /     2           2   \\
cos (x)*\- cos (x) - 9*sin (x)*\- 25*cos (x) + 56*sin (x)/ + 27*cos (x)*sin (x) + 27*cos (x)*\- cos (x) + 8*sin (x)//

$$\left(27 \left(8 \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \cos^{2}{\left(x \right)} - 9 \left(56 \sin^{2}{\left(x \right)} - 25 \cos^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)} + 27 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} - \cos^{4}{\left(x \right)}\right) \cos^{6}{\left(x \right)}$$

Simplify

The graph

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Derivative of sinx*cos^3x^2

The graph:

Piecewise:

The solution