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

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   /   2    \
sin\5*x  + 2/

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

d /   /   2    \\
--\sin\5*x  + 2//
dx

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

Transform

Detail solution

Let $u = 5 x^{2} + 2$ .
The derivative of sine is cosine:

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

$10 x \cos{\left(5 x^{2} + 2 \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

        /   2    \
10*x*cos\5*x  + 2/

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

       /     /       2\       2    /       2\\
-100*x*\3*sin\2 + 5*x / + 10*x *cos\2 + 5*x //

$$- 100 x \left(10 x^{2} \cos{\left(5 x^{2} + 2 \right)} + 3 \sin{\left(5 x^{2} + 2 \right)}\right)$$

Simplify

The graph