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Derivative of:
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Identical expressions
y=sin(5x+ three)^ three
y equally sinus of (5x plus 3) cubed
y equally sinus of (5x plus three) to the power of three
y=sin(5x+3)3
y=sin5x+33
y=sin(5x+3)³
y=sin(5x+3) to the power of 3
y=sin5x+3^3
Similar expressions
y=sin(5x-3)^3

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

Derivative of y=sin(5x+3)^3

The solution

You have entered [src]

   3         
sin (5*x + 3)

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

sin(5*x + 3)^3

Detail solution

Let $u = \sin{\left(5 x + 3 \right)}$ .
Apply the power rule: $u^{3}$ goes to $3 u^{2}$
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 a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $5$
    2. The derivative of the constant $3$ is zero.
    The result is: $5$
  The result of the chain rule is:
  
  $5 \cos{\left(5 x + 3 \right)}$
The result of the chain rule is:

$15 \sin^{2}{\left(5 x + 3 \right)} \cos{\left(5 x + 3 \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      2                      
15*sin (5*x + 3)*cos(5*x + 3)

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

Simplify

The second derivative [src]

   /     2                 2         \             
75*\- sin (3 + 5*x) + 2*cos (3 + 5*x)/*sin(3 + 5*x)

$$75 \left(- \sin^{2}{\left(5 x + 3 \right)} + 2 \cos^{2}{\left(5 x + 3 \right)}\right) \sin{\left(5 x + 3 \right)}$$

Simplify

The third derivative [src]

    /       2                 2         \             
375*\- 7*sin (3 + 5*x) + 2*cos (3 + 5*x)/*cos(3 + 5*x)

$$375 \left(- 7 \sin^{2}{\left(5 x + 3 \right)} + 2 \cos^{2}{\left(5 x + 3 \right)}\right) \cos{\left(5 x + 3 \right)}$$

Simplify

The graph

Derivative of y=sin(5x+3)^3