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Derivative of y=sin^2(x^5)
Identical expressions
y=sin^ two (x^ five)
y equally sinus of squared (x to the power of 5)
y equally sinus of to the power of two (x to the power of five)
y=sin2(x5)
y=sin2x5
y=sin²(x⁵)
y=sin to the power of 2(x to the power of 5)
y=sin^2x^5

The derivative of the function/ y=sin^2(x^5)

Derivative of y=sin^2(x^5)

The solution

You have entered [src]

   2/ 5\
sin \x /

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

sin(x^5)^2

Detail solution

Let $u = \sin{\left(x^{5} \right)}$ .
Apply the power rule: $u^{2}$ goes to $2 u$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x^{5} \right)}$ :
1. Let $u = x^{5}$ .
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 $\frac{d}{d x} x^{5}$ :
  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 result of the chain rule is:

$10 x^{4} \sin{\left(x^{5} \right)} \cos{\left(x^{5} \right)}$
Now simplify:

$5 x^{4} \sin{\left(2 x^{5} \right)}$

The answer is:

$5 x^{4} \sin{\left(2 x^{5} \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    4    / 5\    / 5\
10*x *cos\x /*sin\x /

$$10 x^{4} \sin{\left(x^{5} \right)} \cos{\left(x^{5} \right)}$$

Simplify

The second derivative [src]

    3 /     5    2/ 5\        / 5\    / 5\      5    2/ 5\\
10*x *\- 5*x *sin \x / + 4*cos\x /*sin\x / + 5*x *cos \x //

$$10 x^{3} \left(- 5 x^{5} \sin^{2}{\left(x^{5} \right)} + 5 x^{5} \cos^{2}{\left(x^{5} \right)} + 4 \sin{\left(x^{5} \right)} \cos{\left(x^{5} \right)}\right)$$

Simplify

The third derivative [src]

    2 /      5    2/ 5\        / 5\    / 5\       5    2/ 5\       10    / 5\    / 5\\
40*x *\- 15*x *sin \x / + 3*cos\x /*sin\x / + 15*x *cos \x / - 25*x  *cos\x /*sin\x //

$$40 x^{2} \left(- 25 x^{10} \sin{\left(x^{5} \right)} \cos{\left(x^{5} \right)} - 15 x^{5} \sin^{2}{\left(x^{5} \right)} + 15 x^{5} \cos^{2}{\left(x^{5} \right)} + 3 \sin{\left(x^{5} \right)} \cos{\left(x^{5} \right)}\right)$$

Simplify

The graph

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Identical expressions

Derivative of y=sin^2(x^5)

The graph:

Piecewise:

The solution