Mister Exam

Lang:

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

You have entered [src]

                 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)

Plot the graph f'(x)

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