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  • Identical expressions

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The derivative of the function/ 3a*sin(t^2)*cos(t)

Derivative of 3a*sin(t^2)*cos(t)

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
       / 2\       
3*a*sin\t /*cos(t)
3asin⁡(t2)cos⁡(t)3 a \sin{\left(t^{2} \right)} \cos{\left(t \right)} 
((3*a)*sin(t^2))*cos(t)
Detail solution

  1. Apply the product rule:

    ddtf(t)g(t)=f(t)ddtg(t)+g(t)ddtf(t)\frac{d}{d t} f{\left(t \right)} g{\left(t \right)} = f{\left(t \right)} \frac{d}{d t} g{\left(t \right)} + g{\left(t \right)} \frac{d}{d t} f{\left(t \right)}

    f(t)=3asin⁡(t2)f{\left(t \right)} = 3 a \sin{\left(t^{2} \right)}; to find ddtf(t)\frac{d}{d t} f{\left(t \right)}:

    1. The derivative of a constant times a function is the constant times the derivative of the function.

      1. Let u=t2u = t^{2}.

      2. The derivative of sine is cosine:

        ddusin⁡(u)=cos⁡(u)\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}

      3. Then, apply the chain rule. Multiply by ddtt2\frac{d}{d t} t^{2}:

        1. Apply the power rule: t2t^{2} goes to 2t2 t

        The result of the chain rule is:

        2tcos⁡(t2)2 t \cos{\left(t^{2} \right)}

      So, the result is: 6atcos⁡(t2)6 a t \cos{\left(t^{2} \right)}

    g(t)=cos⁡(t)g{\left(t \right)} = \cos{\left(t \right)}; to find ddtg(t)\frac{d}{d t} g{\left(t \right)}:

    1. The derivative of cosine is negative sine:

      ddtcos⁡(t)=−sin⁡(t)\frac{d}{d t} \cos{\left(t \right)} = - \sin{\left(t \right)}

    The result is: 6atcos⁡(t)cos⁡(t2)−3asin⁡(t)sin⁡(t2)6 a t \cos{\left(t \right)} \cos{\left(t^{2} \right)} - 3 a \sin{\left(t \right)} \sin{\left(t^{2} \right)}

  2. Now simplify:

    3a(2tcos⁡(t)cos⁡(t2)−sin⁡(t)sin⁡(t2))3 a \left(2 t \cos{\left(t \right)} \cos{\left(t^{2} \right)} - \sin{\left(t \right)} \sin{\left(t^{2} \right)}\right)

The answer is:

3a(2tcos⁡(t)cos⁡(t2)−sin⁡(t)sin⁡(t2))3 a \left(2 t \cos{\left(t \right)} \cos{\left(t^{2} \right)} - \sin{\left(t \right)} \sin{\left(t^{2} \right)}\right)

The first derivative [src] 
                / 2\                   / 2\
- 3*a*sin(t)*sin\t / + 6*a*t*cos(t)*cos\t /
6atcos⁡(t)cos⁡(t2)−3asin⁡(t)sin⁡(t2)6 a t \cos{\left(t \right)} \cos{\left(t^{2} \right)} - 3 a \sin{\left(t \right)} \sin{\left(t^{2} \right)}
Simplify
The second derivative [src] 
     /          / 2\     /     / 2\      2    / 2\\                 / 2\       \
-3*a*\cos(t)*sin\t / + 2*\- cos\t / + 2*t *sin\t //*cos(t) + 4*t*cos\t /*sin(t)/
−3a(4tsin⁡(t)cos⁡(t2)+2(2t2sin⁡(t2)−cos⁡(t2))cos⁡(t)+sin⁡(t2)cos⁡(t))- 3 a \left(4 t \sin{\left(t \right)} \cos{\left(t^{2} \right)} + 2 \left(2 t^{2} \sin{\left(t^{2} \right)} - \cos{\left(t^{2} \right)}\right) \cos{\left(t \right)} + \sin{\left(t^{2} \right)} \cos{\left(t \right)}\right)
Simplify
The third derivative [src] 
    /          / 2\     /     / 2\      2    / 2\\                        / 2\       /     / 2\      2    / 2\\       \
3*a*\sin(t)*sin\t / + 6*\- cos\t / + 2*t *sin\t //*sin(t) - 6*t*cos(t)*cos\t / - 4*t*\3*sin\t / + 2*t *cos\t //*cos(t)/
3a(−4t(2t2cos⁡(t2)+3sin⁡(t2))cos⁡(t)−6tcos⁡(t)cos⁡(t2)+6(2t2sin⁡(t2)−cos⁡(t2))sin⁡(t)+sin⁡(t)sin⁡(t2))3 a \left(- 4 t \left(2 t^{2} \cos{\left(t^{2} \right)} + 3 \sin{\left(t^{2} \right)}\right) \cos{\left(t \right)} - 6 t \cos{\left(t \right)} \cos{\left(t^{2} \right)} + 6 \left(2 t^{2} \sin{\left(t^{2} \right)} - \cos{\left(t^{2} \right)}\right) \sin{\left(t \right)} + \sin{\left(t \right)} \sin{\left(t^{2} \right)}\right)
Simplify
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