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

How do you find the derivative of the function R=3W2.πR=3{{W}^{2.\pi }} at W=7W=7 ?

Explanation

Solution

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 xx is a variable and yy is another variable ,then the rate of change of xx with respect yyis given by dydx\dfrac{dy}{dx} . Here , we have to differentiate the entire equation with respect to WW but not with respect to xx.

Complete step by step solution:
Since we are given an equation which is clearly a function of WW , we have to differentiate the entire equation with respect to WW. After doing that, we have substituted the value of WW to find the value of RR at that position.
We are basically finding the slope of the equation R=3W2.πR=3{{W}^{2.\pi }}at W=7W=7.
Before proceeding, we must know the derivative of xn{{x}^{n}} , where nn is any constant .
⇒d(xn)dx=nxn−1\Rightarrow \dfrac{d\left( {{x}^{n}} \right)}{dx}=n{{x}^{n-1}} .
x,nx,n are not related to the question whatsoever. They are just a form of representation.
Now let us differentiate our function with respect to WW.
Upon doing so, we get the following :
⇒R=3W2.π ⇒d(R)dW=3(2π)W2π−1 ⇒d(R)dW=6πW2π−1 \begin{aligned} & \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{aligned}
Now let us substitute W=7W=7.
Upon doing so, we get the following :
⇒R=3W2.π ⇒d(R)dW=3(2π)W2π−1 ⇒d(R)dW=6πW2π−1 ⇒d(R)dW=6π(7)2π−1 \begin{aligned} & \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{aligned}
∴\therefore Hence, the derivative of the function R=3W2.πR=3{{W}^{2.\pi }} at W=7W=7 is 6π(7)2π−16\pi {{\left( 7 \right)}^{2\pi -1}}.

Note: We don’t have to solve further since π\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 xx.