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Identical expressions
y=(sin3x+5x^ two)^ two
y equally ( sinus of 3x plus 5x squared ) squared
y equally ( sinus of 3x plus 5x to the power of two) to the power of two
y=(sin3x+5x2)2
y=sin3x+5x22
y=(sin3x+5x²)²
y=(sin3x+5x to the power of 2) to the power of 2
y=sin3x+5x^2^2
Similar expressions
y=(sin3x-5x^2)^2

The derivative of the function/ y=(sin3x+5x^2)^2

Derivative of y=(sin3x+5x^2)^2

The solution

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                 2
/              2\ 
\sin(3*x) + 5*x /

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

  /                 2\
d |/              2\ |
--\\sin(3*x) + 5*x / /
dx

$$\frac{d}{d x} \left(5 x^{2} + \sin{\left(3 x \right)}\right)^{2}$$

Transform

Detail solution

Let $u = 5 x^{2} + \sin{\left(3 x \right)}$ .
Apply the power rule: $u^{2}$ goes to $2 u$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(5 x^{2} + \sin{\left(3 x \right)}\right)$ :
1. Differentiate $5 x^{2} + \sin{\left(3 x \right)}$ term by term:
  1. Let $u = 3 x$ .
  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 $\frac{d}{d x} 3 x$ :
    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: $3$
    The result of the chain rule is:
    
    $3 \cos{\left(3 x \right)}$
  4. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    So, the result is: $10 x$
  The result is: $10 x + 3 \cos{\left(3 x \right)}$
The result of the chain rule is:

$\left(10 x + 3 \cos{\left(3 x \right)}\right) \left(10 x^{2} + 2 \sin{\left(3 x \right)}\right)$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

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The first derivative [src]

                    /              2\
(6*cos(3*x) + 20*x)*\sin(3*x) + 5*x /

$$\left(20 x + 6 \cos{\left(3 x \right)}\right) \left(5 x^{2} + \sin{\left(3 x \right)}\right)$$

Simplify

The second derivative [src]

  /                   2                      /   2           \\
2*\(3*cos(3*x) + 10*x)  - (-10 + 9*sin(3*x))*\5*x  + sin(3*x)//

$$2 \left(\left(10 x + 3 \cos{\left(3 x \right)}\right)^{2} - \left(5 x^{2} + \sin{\left(3 x \right)}\right) \left(9 \sin{\left(3 x \right)} - 10\right)\right)$$

Simplify

The third derivative [src]

   /                                           /   2           \         \
-6*\(-10 + 9*sin(3*x))*(3*cos(3*x) + 10*x) + 9*\5*x  + sin(3*x)/*cos(3*x)/

$$- 6 \left(\left(10 x + 3 \cos{\left(3 x \right)}\right) \left(9 \sin{\left(3 x \right)} - 10\right) + 9 \cdot \left(5 x^{2} + \sin{\left(3 x \right)}\right) \cos{\left(3 x \right)}\right)$$

Simplify

The graph

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Derivative of y=(sin3x+5x^2)^2

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The solution