Derivative of y=sin(x^5)

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   / 5\
sin\x /

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

sin(x^5)

Detail solution

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

$\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} x^{5}$ :
1. Apply the power rule: $x^{5}$ goes to $5 x^{4}$
The result of the chain rule is:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   4    / 5\
5*x *cos\x /

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

Simplify

The second derivative [src]

   3 /     / 5\      5    / 5\\
5*x *\4*cos\x / - 5*x *sin\x //

$$5 x^{3} \left(- 5 x^{5} \sin{\left(x^{5} \right)} + 4 \cos{\left(x^{5} \right)}\right)$$

Simplify

The third derivative [src]

   2 /      / 5\       5    / 5\       10    / 5\\
5*x *\12*cos\x / - 60*x *sin\x / - 25*x  *cos\x //

$$5 x^{2} \left(- 25 x^{10} \cos{\left(x^{5} \right)} - 60 x^{5} \sin{\left(x^{5} \right)} + 12 \cos{\left(x^{5} \right)}\right)$$

Simplify

The graph