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Derivative of:
Derivative of asin(t)
Derivative of 7/x^4
Derivative of -(x^2+25)/x
Derivative of (x^2+1)^4
Identical expressions
a*sin(t)^ two
a multiply by sinus of (t) squared
a multiply by sinus of (t) to the power of two
a*sin(t)2
a*sint2
a*sin(t)²
a*sin(t) to the power of 2
asin(t)^2
asin(t)2
asint2
asint^2

The derivative of the function/ a*sin(t)^2

Derivative of a*sin(t)^2

The solution

You have entered [src]

     2   
a*sin (t)

$$a \sin^{2}{\left(t \right)}$$

a*sin(t)^2

Detail solution

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

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

The answer is:

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

The first derivative [src]

2*a*cos(t)*sin(t)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify