Derivative of sin(t)^3

The calculator will find the derivative of

\sin^{3}{\left(t \right)}

, with steps shown.

Your Input

Find $\frac{d}{dt} \left(\sin^{3}{\left(t \right)}\right)$ .

Solution

The function $\sin^{3}{\left(t \right)}$ is the composition $f{\left(g{\left(t \right)} \right)}$ of two functions $f{\left(u \right)} = u^{3}$ and $g{\left(t \right)} = \sin{\left(t \right)}$ .

Apply the chain rule $\frac{d}{dt} \left(f{\left(g{\left(t \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dt} \left(g{\left(t \right)}\right)$ :

{\color{red}\left(\frac{d}{dt} \left(\sin^{3}{\left(t \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(u^{3}\right) \frac{d}{dt} \left(\sin{\left(t \right)}\right)\right)}

Apply the power rule $\frac{d}{du} \left(u^{n}\right) = n u^{n - 1}$ with $n = 3$ :

{\color{red}\left(\frac{d}{du} \left(u^{3}\right)\right)} \frac{d}{dt} \left(\sin{\left(t \right)}\right) = {\color{red}\left(3 u^{2}\right)} \frac{d}{dt} \left(\sin{\left(t \right)}\right)

Return to the old variable:

3 {\color{red}\left(u\right)}^{2} \frac{d}{dt} \left(\sin{\left(t \right)}\right) = 3 {\color{red}\left(\sin{\left(t \right)}\right)}^{2} \frac{d}{dt} \left(\sin{\left(t \right)}\right)

The derivative of the sine is $\frac{d}{dt} \left(\sin{\left(t \right)}\right) = \cos{\left(t \right)}$ :

3 \sin^{2}{\left(t \right)} {\color{red}\left(\frac{d}{dt} \left(\sin{\left(t \right)}\right)\right)} = 3 \sin^{2}{\left(t \right)} {\color{red}\left(\cos{\left(t \right)}\right)}

Thus, $\frac{d}{dt} \left(\sin^{3}{\left(t \right)}\right) = 3 \sin^{2}{\left(t \right)} \cos{\left(t \right)}$ .

Answer

$\frac{d}{dt} \left(\sin^{3}{\left(t \right)}\right) = 3 \sin^{2}{\left(t \right)} \cos{\left(t \right)}$ A

Derivative of sin⁡3(t)\sin^{3}{\left(t \right)}sin3(t)

Your Input

Solution

Answer

Derivative of $\sin^{3}{\left(t \right)}$