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Derivative of y=sin(x^2-x+1)

You have entered [src]

   / 2        \
sin\x  - x + 1/

$$\sin{\left(x^{2} - x + 1 \right)}$$

d /   / 2        \\
--\sin\x  - x + 1//
dx

$$\frac{d}{d x} \sin{\left(x^{2} - x + 1 \right)}$$

Transform

Detail solution

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

$\left(2 x - 1\right) \cos{\left(x^{2} - x + 1 \right)}$
Now simplify:

$\left(2 x - 1\right) \cos{\left(x^{2} - x + 1 \right)}$

The answer is:

$\left(2 x - 1\right) \cos{\left(x^{2} - x + 1 \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

              / 2        \
(-1 + 2*x)*cos\x  - x + 1/

$$\left(2 x - 1\right) \cos{\left(x^{2} - x + 1 \right)}$$

Simplify

The second derivative [src]

     /     2    \             2    /     2    \
2*cos\1 + x  - x/ - (-1 + 2*x) *sin\1 + x  - x/

$$- \left(2 x - 1\right)^{2} \sin{\left(x^{2} - x + 1 \right)} + 2 \cos{\left(x^{2} - x + 1 \right)}$$

Simplify

The third derivative [src]

            /     /     2    \             2    /     2    \\
-(-1 + 2*x)*\6*sin\1 + x  - x/ + (-1 + 2*x) *cos\1 + x  - x//

$$- \left(2 x - 1\right) \left(\left(2 x - 1\right)^{2} \cos{\left(x^{2} - x + 1 \right)} + 6 \sin{\left(x^{2} - x + 1 \right)}\right)$$

Simplify

The graph