Question

How do you find the derivative of $s=t\sin t$?

Answer 1

Consider ‘s’ in the L.H.S as a function of t and differentiate both the sides with respect to the variable t. Consider ‘s’ as the product of an algebraic function and a trigonometric function. Now, apply the product rule of differentiation given as: - $\dfrac{d\left( u\times v \right)}{dt}=u\dfrac{dv}{dt}+v\dfrac{du}{dt}$. Here, consider, u = t and $v=\sin t$. Use the formula: - $\dfrac{d\sin t}{dt}=\cos t$ to simplify the derivative and get the answer.

Complete step by step solution:
Here, we have been provided with the function $s=t\sin t$ and we are asked to differentiate it. Here we are going to use the product rule of differentiation to get the answer.
$\because s=t\sin t$
Clearly, we can see that we have ‘s’ as a function of t. Now, we can assume the given function as the product of an algebraic function (t) and a trigonometric function $\left( \sin t \right)$. So, we have,
$\Rightarrow s=t\times \sin t$
Let us assume t as ‘u’ and $\sin t$ as ‘v’. So, we have,
$\Rightarrow s=u\times v$
Differentiating both the sides with respect to t, we get,
$\Rightarrow \dfrac{ds}{dt}=\dfrac{d\left( u\times v \right)}{dt}$
Now, applying the product rule of differentiation given as: - $\dfrac{d\left( u\times v \right)}{dt}=u\dfrac{dv}{dt}+v\dfrac{du}{dt}$, we get,
$\Rightarrow \dfrac{ds}{dt}=\left[ u\dfrac{dv}{dt}+v\dfrac{du}{dt} \right]$
Substituting the assumed values of u and v, we get,
$\Rightarrow \dfrac{ds}{dt}=\left[ t\dfrac{d\sin t}{dt}+t\dfrac{dt}{dt} \right]$
We know that $\dfrac{d\sin t}{dt}=\cos t$, so we have,

& \Rightarrow \dfrac{ds}{dt}=\left[ t\cos t+\sin t\times 1 \right] \\\ & \Rightarrow \dfrac{ds}{dt}=\left( t\cos t+\sin t \right) \\\ \end{aligned}$$ Hence, the above relation is our answer. **Note:** One may note that whenever we are asked to differentiate a product of two or more functions we apply the product rule. You must remember all the basic rules and formulas of differentiation like: - the product rule, chain rule, $$\dfrac{u}{v}$$ rule etc. as they are frequently used in both differential and integral calculus. Remember the derivatives of some common functions like: algebraic functions, trigonometric functions, logarithmic functions, exponential functions etc. as we may be asked to find the derivative of the product of any two of these listed functions.