Derivative of y=sin(5x²+4x+3)

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sin\5*x  + 4*x + 3/

$$\sin{\left(\left(5 x^{2} + 4 x\right) + 3 \right)}$$

sin(5*x^2 + 4*x + 3)

Detail solution

Let $u = \left(5 x^{2} + 4 x\right) + 3$ .
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 $\frac{d}{d x} \left(\left(5 x^{2} + 4 x\right) + 3\right)$ :
1. Differentiate $\left(5 x^{2} + 4 x\right) + 3$ term by term:
  1. Differentiate $5 x^{2} + 4 x$ 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 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: $4$
    The result is: $10 x + 4$
  2. The derivative of the constant $3$ is zero.
  The result is: $10 x + 4$
The result of the chain rule is:

$\left(10 x + 4\right) \cos{\left(\left(5 x^{2} + 4 x\right) + 3 \right)}$
Now simplify:

$\left(10 x + 4\right) \cos{\left(5 x^{2} + 4 x + 3 \right)}$

The answer is:

$\left(10 x + 4\right) \cos{\left(5 x^{2} + 4 x + 3 \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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(4 + 10*x)*cos\5*x  + 4*x + 3/

$$\left(10 x + 4\right) \cos{\left(\left(5 x^{2} + 4 x\right) + 3 \right)}$$

Simplify

The second derivative [src]

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2*\5*cos\3 + 4*x + 5*x / - 2*(2 + 5*x) *sin\3 + 4*x + 5*x //

$$2 \left(- 2 \left(5 x + 2\right)^{2} \sin{\left(5 x^{2} + 4 x + 3 \right)} + 5 \cos{\left(5 x^{2} + 4 x + 3 \right)}\right)$$

Simplify

The third derivative [src]

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-4*(2 + 5*x)*\15*sin\3 + 4*x + 5*x / + 2*(2 + 5*x) *cos\3 + 4*x + 5*x //

$$- 4 \left(5 x + 2\right) \left(2 \left(5 x + 2\right)^{2} \cos{\left(5 x^{2} + 4 x + 3 \right)} + 15 \sin{\left(5 x^{2} + 4 x + 3 \right)}\right)$$

Simplify

The graph