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Similar expressions
sin^8(3x^3+5x)

The derivative of the function/ sin^8(3x^3-5x)

Derivative of sin^8(3x^3-5x)

The solution

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   8/   3      \
sin \3*x  - 5*x/

$$\sin^{8}{\left(3 x^{3} - 5 x \right)}$$

d /   8/   3      \\
--\sin \3*x  - 5*x//
dx

$$\frac{d}{d x} \sin^{8}{\left(3 x^{3} - 5 x \right)}$$

Transform

Detail solution

Let $u = \sin{\left(3 x^{3} - 5 x \right)}$ .
Apply the power rule: $u^{8}$ goes to $8 u^{7}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(3 x^{3} - 5 x \right)}$ :
1. Let $u = 3 x^{3} - 5 x$ .
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 $\frac{d}{d x} \left(3 x^{3} - 5 x\right)$ :
  1. Differentiate $3 x^{3} - 5 x$ 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: $x^{3}$ goes to $3 x^{2}$
      So, the result is: $9 x^{2}$
    2. The derivative of a constant times a function is the constant times the derivative of the function.
      1. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $5$
      So, the result is: $-5$
    The result is: $9 x^{2} - 5$
  The result of the chain rule is:
  
  $\left(9 x^{2} - 5\right) \cos{\left(3 x^{3} - 5 x \right)}$
The result of the chain rule is:

$8 \cdot \left(9 x^{2} - 5\right) \sin^{7}{\left(3 x^{3} - 5 x \right)} \cos{\left(3 x^{3} - 5 x \right)}$
Now simplify:

$\left(72 x^{2} - 40\right) \sin^{7}{\left(x \left(3 x^{2} - 5\right) \right)} \cos{\left(x \left(3 x^{2} - 5\right) \right)}$

The answer is:

$\left(72 x^{2} - 40\right) \sin^{7}{\left(x \left(3 x^{2} - 5\right) \right)} \cos{\left(x \left(3 x^{2} - 5\right) \right)}$

The graph

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Plot the graph f'(x)

The first derivative [src]

     7/   3      \ /        2\    /   3      \
8*sin \3*x  - 5*x/*\-5 + 9*x /*cos\3*x  - 5*x/

$$8 \cdot \left(9 x^{2} - 5\right) \sin^{7}{\left(3 x^{3} - 5 x \right)} \cos{\left(3 x^{3} - 5 x \right)}$$

Simplify

The second derivative [src]

                      /             2                                    2                                                                 \
     6/  /        2\\ |  /        2\     2/  /        2\\     /        2\     2/  /        2\\           /  /        2\\    /  /        2\\|
8*sin \x*\-5 + 3*x //*\- \-5 + 9*x / *sin \x*\-5 + 3*x // + 7*\-5 + 9*x / *cos \x*\-5 + 3*x // + 18*x*cos\x*\-5 + 3*x //*sin\x*\-5 + 3*x ///

$$8 \cdot \left(- \left(9 x^{2} - 5\right)^{2} \sin^{2}{\left(x \left(3 x^{2} - 5\right) \right)} + 7 \left(9 x^{2} - 5\right)^{2} \cos^{2}{\left(x \left(3 x^{2} - 5\right) \right)} + 18 x \sin{\left(x \left(3 x^{2} - 5\right) \right)} \cos{\left(x \left(3 x^{2} - 5\right) \right)}\right) \sin^{6}{\left(x \left(3 x^{2} - 5\right) \right)}$$

Simplify

The third derivative [src]

                       /                                                         3                                                                            3                                                                                                  \
      5/  /        2\\ |     2/  /        2\\    /  /        2\\      /        2\     3/  /        2\\           3/  /        2\\ /        2\      /        2\     2/  /        2\\    /  /        2\\            2/  /        2\\ /        2\    /  /        2\\|
16*sin \x*\-5 + 3*x //*\9*sin \x*\-5 + 3*x //*cos\x*\-5 + 3*x // + 21*\-5 + 9*x / *cos \x*\-5 + 3*x // - 27*x*sin \x*\-5 + 3*x //*\-5 + 9*x / - 11*\-5 + 9*x / *sin \x*\-5 + 3*x //*cos\x*\-5 + 3*x // + 189*x*cos \x*\-5 + 3*x //*\-5 + 9*x /*sin\x*\-5 + 3*x ///

$$16 \left(- 11 \left(9 x^{2} - 5\right)^{3} \sin^{2}{\left(x \left(3 x^{2} - 5\right) \right)} \cos{\left(x \left(3 x^{2} - 5\right) \right)} + 21 \left(9 x^{2} - 5\right)^{3} \cos^{3}{\left(x \left(3 x^{2} - 5\right) \right)} - 27 x \left(9 x^{2} - 5\right) \sin^{3}{\left(x \left(3 x^{2} - 5\right) \right)} + 189 x \left(9 x^{2} - 5\right) \sin{\left(x \left(3 x^{2} - 5\right) \right)} \cos^{2}{\left(x \left(3 x^{2} - 5\right) \right)} + 9 \sin^{2}{\left(x \left(3 x^{2} - 5\right) \right)} \cos{\left(x \left(3 x^{2} - 5\right) \right)}\right) \sin^{5}{\left(x \left(3 x^{2} - 5\right) \right)}$$

Simplify

The graph

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Derivative of sin^8(3x^3-5x)

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The solution