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Derivative of cos(x^3-5)

You have entered [src]

   / 3    \
cos\x  - 5/

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

cos(x^3 - 5)

Detail solution

Let $u = x^{3} - 5$ .
The derivative of cosine is negative sine:

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

$- 3 x^{2} \sin{\left(x^{3} - 5 \right)}$
Now simplify:

$- 3 x^{2} \sin{\left(x^{3} - 5 \right)}$

The answer is:

$- 3 x^{2} \sin{\left(x^{3} - 5 \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    2    / 3    \
-3*x *sin\x  - 5/

$$- 3 x^{2} \sin{\left(x^{3} - 5 \right)}$$

Simplify

The second derivative [src]

     /     /      3\      3    /      3\\
-3*x*\2*sin\-5 + x / + 3*x *cos\-5 + x //

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

Simplify

The third derivative [src]

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

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

Simplify

The graph