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Identical expressions
a*sinx^ two + one
a multiply by sinus of x squared plus 1
a multiply by sinus of x to the power of two plus one
a*sinx2+1
a*sinx²+1
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Similar expressions
a*sinx^2-1

The derivative of the function/ a*sinx^2+1

Derivative of a*sinx^2+1

The solution

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     2       
a*sin (x) + 1

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

d /     2       \
--\a*sin (x) + 1/
dx

$$\frac{\partial}{\partial x} \left(a \sin^{2}{\left(x \right)} + 1\right)$$

Transform

Detail solution

Differentiate $a \sin^{2}{\left(x \right)} + 1$ term by term:
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = \sin{\left(x \right)}$ .
  2. Apply the power rule: $u^{2}$ goes to $2 u$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x \right)}$ :
    1. The derivative of sine is cosine:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
    The result of the chain rule is:
    
    $2 \sin{\left(x \right)} \cos{\left(x \right)}$
  So, the result is: $2 a \sin{\left(x \right)} \cos{\left(x \right)}$
2. The derivative of the constant $1$ is zero.
The result is: $2 a \sin{\left(x \right)} \cos{\left(x \right)}$
Now simplify:

$a \sin{\left(2 x \right)}$

The answer is:

$a \sin{\left(2 x \right)}$

The first derivative [src]

2*a*cos(x)*sin(x)

$$2 a \sin{\left(x \right)} \cos{\left(x \right)}$$

Simplify

The second derivative [src]

    /   2         2   \
2*a*\cos (x) - sin (x)/

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

Simplify

The third derivative [src]

-8*a*cos(x)*sin(x)

$$- 8 a \sin{\left(x \right)} \cos{\left(x \right)}$$

Simplify