Derivative of y=sin³(x²+2x)

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   3/ 2      \
sin \x  + 2*x/

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

d /   3/ 2      \\
--\sin \x  + 2*x//
dx

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

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Detail solution

Let $u = \sin{\left(x^{2} + 2 x \right)}$ .
Apply the power rule: $u^{3}$ goes to $3 u^{2}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x^{2} + 2 x \right)}$ :
1. Let $u = x^{2} + 2 x$ .
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 $\frac{d}{d x} \left(x^{2} + 2 x\right)$ :
  1. Differentiate $x^{2} + 2 x$ term by term:
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    2. 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: $2$
    The result is: $2 x + 2$
  The result of the chain rule is:
  
  $\left(2 x + 2\right) \cos{\left(x^{2} + 2 x \right)}$
The result of the chain rule is:

$3 \cdot \left(2 x + 2\right) \sin^{2}{\left(x^{2} + 2 x \right)} \cos{\left(x^{2} + 2 x \right)}$
Now simplify:

$\left(6 x + 6\right) \sin^{2}{\left(x \left(x + 2\right) \right)} \cos{\left(x \left(x + 2\right) \right)}$

The answer is:

$\left(6 x + 6\right) \sin^{2}{\left(x \left(x + 2\right) \right)} \cos{\left(x \left(x + 2\right) \right)}$

The graph

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The first derivative [src]

     2/ 2      \              / 2      \
3*sin \x  + 2*x/*(2 + 2*x)*cos\x  + 2*x/

$$3 \cdot \left(2 x + 2\right) \sin^{2}{\left(x^{2} + 2 x \right)} \cos{\left(x^{2} + 2 x \right)}$$

Simplify

The second derivative [src]

  /                                         2    2                       2    2           \               
6*\cos(x*(2 + x))*sin(x*(2 + x)) - 2*(1 + x) *sin (x*(2 + x)) + 4*(1 + x) *cos (x*(2 + x))/*sin(x*(2 + x))

$$6 \left(- 2 \left(x + 1\right)^{2} \sin^{2}{\left(x \left(x + 2\right) \right)} + 4 \left(x + 1\right)^{2} \cos^{2}{\left(x \left(x + 2\right) \right)} + \sin{\left(x \left(x + 2\right) \right)} \cos{\left(x \left(x + 2\right) \right)}\right) \sin{\left(x \left(x + 2\right) \right)}$$

Simplify

The third derivative [src]

           /       3                       2    3                   2                                       2    2                          \
12*(1 + x)*\- 3*sin (x*(2 + x)) + 4*(1 + x) *cos (x*(2 + x)) + 6*cos (x*(2 + x))*sin(x*(2 + x)) - 14*(1 + x) *sin (x*(2 + x))*cos(x*(2 + x))/

$$12 \left(x + 1\right) \left(- 14 \left(x + 1\right)^{2} \sin^{2}{\left(x \left(x + 2\right) \right)} \cos{\left(x \left(x + 2\right) \right)} + 4 \left(x + 1\right)^{2} \cos^{3}{\left(x \left(x + 2\right) \right)} - 3 \sin^{3}{\left(x \left(x + 2\right) \right)} + 6 \sin{\left(x \left(x + 2\right) \right)} \cos^{2}{\left(x \left(x + 2\right) \right)}\right)$$

Simplify

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Derivative of y=sin³(x²+2x)

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