Find 𝑓 prime of four, if 𝑓 of 𝑥 is equal to eight 𝑒 to the power of six 𝑥.
𝑓 prime of four means the first derivative of our function 𝑓 evaluated when 𝑥 is equal to four. We notice that our function 𝑓 of 𝑥 is an exponential function. So we need to recall the standard rules for differentiating these. We recall that the derivative with respect to 𝑥 of 𝑒 to the power of some function 𝑔 of 𝑥 is equal to 𝑔 prime of 𝑥 multiplied by 𝑒 to the power of 𝑔 of 𝑥. This is a direct application of the chain rule.
Here, our function 𝑔 of 𝑥 is six 𝑥. And applying the power rule of differentiation, we find that 𝑔 prime of 𝑥 is equal to six. The factor of eight is just a multiplicative constant. And we know that when we differentiate, we just need to multiply a derivative by eight. So we have that 𝑓 prime of 𝑥 is equal to eight multiplied by 𝑔 prime of 𝑥 — that’s six — multiplied by 𝑒 to the power of 𝑔 of 𝑥 — that’s 𝑒 to the power of six 𝑥. Now, we could simplify and write this as 48𝑒 to the power of six 𝑥 before we substitute 𝑥 equals four to evaluate its derivative.
But one thing that’s nice about differentiating exponential functions is that we can express their derivatives in terms of the original function, if we wish. Our original function 𝑓 of 𝑥 was eight 𝑒 to the power of six 𝑥. So if we wanted, we could express 𝑓 prime of 𝑥 as six multiplied by 𝑓 of 𝑥. When we find 𝑓 prime of four then, we can give our answer in terms of the original function. In this case, substituting 𝑥 equals four gives that 𝑓 prime of four is equal to six multiplied by 𝑓 of four.
Now, whether you answer the question this way or whether you substitute 𝑥 equals four into the explicit expression for 𝑓 prime of 𝑥, we’ll depend on the requirements of the particular question we’re answering. In this case, we’ll give our answer as 𝑓 prime of four is equal to six 𝑓 of four.