Question

How do you find the derivative of the function $R=3{{W}^{2.\pi }}$ at $W=7$ ?

Answer 1

Hint: Differentiation is a process where we find the instantaneous rate of change in function based on one of its variables. It is a method finding the derivative of a function. The most common example is the rate of change of displacement with respect to time, called velocity. If $x$ is a variable and $y$ is another variable ,then the rate of change of $x$ with respect $y$is given by $\dfrac{dy}{dx}$ . Here , we have to differentiate the entire equation with respect to $W$ but not with respect to $x$.

Complete step by step solution:
Since we are given an equation which is clearly a function of $W$ , we have to differentiate the entire equation with respect to $W$. After doing that, we have substituted the value of $W$ to find the value of $R$ at that position.
We are basically finding the slope of the equation $R=3{{W}^{2.\pi }}$at $W=7$.
Before proceeding, we must know the derivative of ${{x}^{n}}$ , where $n$ is any constant .
\[\Rightarrow \dfrac{d\left( {{x}^{n}} \right)}{dx}=n{{x}^{n-1}}\] .
$x,n$ are not related to the question whatsoever. They are just a form of representation.
Now let us differentiate our function with respect to $W$.
Upon doing so, we get the following :
$\begin{align}
  & \Rightarrow R=3{{W}^{2.\pi }} \\
 & \Rightarrow \dfrac{d\left( R \right)}{dW}=3\left( 2\pi \right){{W}^{2\pi -1}} \\
 & \Rightarrow \dfrac{d\left( R \right)}{dW}=6\pi {{W}^{2\pi -1}} \\
\end{align}$
Now let us substitute $W=7$.
Upon doing so, we get the following :
$\begin{align}
  & \Rightarrow R=3{{W}^{2.\pi }} \\
 & \Rightarrow \dfrac{d\left( R \right)}{dW}=3\left( 2\pi \right){{W}^{2\pi -1}} \\
 & \Rightarrow \dfrac{d\left( R \right)}{dW}=6\pi {{W}^{2\pi -1}} \\
 & \Rightarrow \dfrac{d\left( R \right)}{dW}=6\pi {{\left( 7 \right)}^{2\pi -1}} \\
\end{align}$
$\therefore $ Hence, the derivative of the function $R=3{{W}^{2.\pi }}$ at $W=7$ is $6\pi {{\left( 7 \right)}^{2\pi -1}}$.

Note: We don’t have to solve further since $\pi $ is non-terminating decimal. There is no perfect value we can round up to so as to solve further. We can leave it as it is. We should remember all the rules of differentiation. We should also remember all the derivatives of all the functions to solve a question quickly in the exam. We should also be able to differentiate different kinds of function. Practice is needed. We should not always blindly differentiate a function with respect to $x$.