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Derivative of y=sin(x^2-3x+5)

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   / 2          \
sin\x  - 3*x + 5/

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

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

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

Transform

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph