Question

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

Answer 1

Hint: 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: - $${\displaystyle \frac{d(u\times v)}{dt}}=u{\displaystyle \frac{dv}{dt}}+v{\displaystyle \frac{du}{dt}}$$. Here, consider, u = t and $$v=\sin t$$. Use the formula: - $${\displaystyle \frac{d\sin t}{dt}}=\cos t$$ 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 $$(\sin t)$$. 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,
$$\Rightarrow {\displaystyle \frac{ds}{dt}}={\displaystyle \frac{d(u\times v)}{dt}}$$
Now, applying the product rule of differentiation given as: - $${\displaystyle \frac{d(u\times v)}{dt}}=u{\displaystyle \frac{dv}{dt}}+v{\displaystyle \frac{du}{dt}}$$, we get,
$$\Rightarrow {\displaystyle \frac{ds}{dt}}=[u{\displaystyle \frac{dv}{dt}}+v{\displaystyle \frac{du}{dt}}]$$
Substituting the assumed values of u and v, we get,
$$\Rightarrow {\displaystyle \frac{ds}{dt}}=[t{\displaystyle \frac{d\sin t}{dt}}+t{\displaystyle \frac{dt}{dt}}]$$
We know that $${\displaystyle \frac{d\sin t}{dt}}=\cos t$$, so we have,
$$\begin{array}{rl}& \Rightarrow {\displaystyle \frac{ds}{dt}}=[t\cos t+\sin t\times 1]\\ & \Rightarrow {\displaystyle \frac{ds}{dt}}=(t\cos t+\sin t)\end{array}$$
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, $${\displaystyle \frac{u}{v}}$$ 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.