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cos^3(2x^2-1)

The derivative of the function/ cos^3(2x^2+1)

Derivative of cos^3(2x^2+1)

The solution

You have entered [src]

   3/   2    \
cos \2*x  + 1/

$$\cos^{3}{\left(2 x^{2} + 1 \right)}$$

d /   3/   2    \\
--\cos \2*x  + 1//
dx

$$\frac{d}{d x} \cos^{3}{\left(2 x^{2} + 1 \right)}$$

Transform

Detail solution

Let $u = \cos{\left(2 x^{2} + 1 \right)}$ .
Apply the power rule: $u^{3}$ goes to $3 u^{2}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(2 x^{2} + 1 \right)}$ :
1. Let $u = 2 x^{2} + 1$ .
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 $\frac{d}{d x} \left(2 x^{2} + 1\right)$ :
  1. Differentiate $2 x^{2} + 1$ 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: $x^{2}$ goes to $2 x$
      So, the result is: $4 x$
    2. The derivative of the constant $1$ is zero.
    The result is: $4 x$
  The result of the chain rule is:
  
  $- 4 x \sin{\left(2 x^{2} + 1 \right)}$
The result of the chain rule is:

$- 12 x \sin{\left(2 x^{2} + 1 \right)} \cos^{2}{\left(2 x^{2} + 1 \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

         2/   2    \    /   2    \
-12*x*cos \2*x  + 1/*sin\2*x  + 1/

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

Simplify

The second derivative [src]

   /     /       2\    /       2\      2    2/       2\      2    2/       2\\    /       2\
12*\- cos\1 + 2*x /*sin\1 + 2*x / - 4*x *cos \1 + 2*x / + 8*x *sin \1 + 2*x //*cos\1 + 2*x /

$$12 \cdot \left(8 x^{2} \sin^{2}{\left(2 x^{2} + 1 \right)} - 4 x^{2} \cos^{2}{\left(2 x^{2} + 1 \right)} - \sin{\left(2 x^{2} + 1 \right)} \cos{\left(2 x^{2} + 1 \right)}\right) \cos{\left(2 x^{2} + 1 \right)}$$

Simplify

3-я производная [src]

     /       3/       2\      2    3/       2\        2/       2\    /       2\       2    2/       2\    /       2\\
48*x*\- 3*cos \1 + 2*x / - 8*x *sin \1 + 2*x / + 6*sin \1 + 2*x /*cos\1 + 2*x / + 28*x *cos \1 + 2*x /*sin\1 + 2*x //

$$48 x \left(- 8 x^{2} \sin^{3}{\left(2 x^{2} + 1 \right)} + 28 x^{2} \sin{\left(2 x^{2} + 1 \right)} \cos^{2}{\left(2 x^{2} + 1 \right)} + 6 \sin^{2}{\left(2 x^{2} + 1 \right)} \cos{\left(2 x^{2} + 1 \right)} - 3 \cos^{3}{\left(2 x^{2} + 1 \right)}\right)$$

Simplify

The third derivative [src]

     /       3/       2\      2    3/       2\        2/       2\    /       2\       2    2/       2\    /       2\\
48*x*\- 3*cos \1 + 2*x / - 8*x *sin \1 + 2*x / + 6*sin \1 + 2*x /*cos\1 + 2*x / + 28*x *cos \1 + 2*x /*sin\1 + 2*x //

Simplify

The graph

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Derivative of cos^3(2x^2+1)

The graph:

Piecewise:

The solution