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Identical expressions
y=sin^ four (five -x^ three)
y equally sinus of to the power of 4(5 minus x cubed )
y equally sinus of to the power of four (five minus x to the power of three)
y=sin4(5-x3)
y=sin45-x3
y=sin⁴(5-x³)
y=sin to the power of 4(5-x to the power of 3)
y=sin^45-x^3
Similar expressions
y=sin^4(5+x^3)

The derivative of the function/ y=sin^4(5-x^3)

Derivative of y=sin^4(5-x^3)

The solution

You have entered [src]

   4/     3\
sin \5 - x /

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

d /   4/     3\\
--\sin \5 - x //
dx

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

Transform

Detail solution

Let $u = \sin{\left(5 - x^{3} \right)}$ .
Apply the power rule: $u^{4}$ goes to $4 u^{3}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(5 - x^{3} \right)}$ :
1. Let $u = 5 - x^{3}$ .
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 $\frac{d}{d x} \left(5 - x^{3}\right)$ :
  1. Differentiate $5 - x^{3}$ term by term:
    1. The derivative of the constant $5$ is zero.
    2. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$
      So, the result is: $- 3 x^{2}$
    The result is: $- 3 x^{2}$
  The result of the chain rule is:
  
  $- 3 x^{2} \cos{\left(x^{3} - 5 \right)}$
The result of the chain rule is:

$- 12 x^{2} \sin^{3}{\left(5 - x^{3} \right)} \cos{\left(x^{3} - 5 \right)}$
Now simplify:

$12 x^{2} \sin^{3}{\left(x^{3} - 5 \right)} \cos{\left(x^{3} - 5 \right)}$

The answer is:

$12 x^{2} \sin^{3}{\left(x^{3} - 5 \right)} \cos{\left(x^{3} - 5 \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     2    3/     3\    /      3\
-12*x *sin \5 - x /*cos\-5 + x /

$$- 12 x^{2} \sin^{3}{\left(- x^{3} + 5 \right)} \cos{\left(x^{3} - 5 \right)}$$

Simplify

The second derivative [src]

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

$$12 x \left(- 3 x^{3} \sin^{2}{\left(x^{3} - 5 \right)} + 9 x^{3} \cos^{2}{\left(x^{3} - 5 \right)} + 2 \sin{\left(x^{3} - 5 \right)} \cos{\left(x^{3} - 5 \right)}\right) \sin^{2}{\left(x^{3} - 5 \right)}$$

Simplify

The third derivative [src]

   /   2/      3\    /      3\      3    3/      3\       6    3/      3\       6    2/      3\    /      3\       3    2/      3\    /      3\\    /      3\
24*\sin \-5 + x /*cos\-5 + x / - 9*x *sin \-5 + x / + 27*x *cos \-5 + x / - 45*x *sin \-5 + x /*cos\-5 + x / + 27*x *cos \-5 + x /*sin\-5 + x //*sin\-5 + x /

$$24 \left(- 45 x^{6} \sin^{2}{\left(x^{3} - 5 \right)} \cos{\left(x^{3} - 5 \right)} + 27 x^{6} \cos^{3}{\left(x^{3} - 5 \right)} - 9 x^{3} \sin^{3}{\left(x^{3} - 5 \right)} + 27 x^{3} \sin{\left(x^{3} - 5 \right)} \cos^{2}{\left(x^{3} - 5 \right)} + \sin^{2}{\left(x^{3} - 5 \right)} \cos{\left(x^{3} - 5 \right)}\right) \sin{\left(x^{3} - 5 \right)}$$

Simplify

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Derivative of y=sin^4(5-x^3)

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The solution