# Question Video: Finding the First Derivative of an Exponential Function Using the Chain Rule at a Point Mathematics • Higher Education

Find πβ²(4), if π(π₯) = 8π^(6π₯).

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### Video Transcript

Find π prime of four, if π of π₯ is equal to eight π to the power of six π₯.

π prime of four means the first derivative of our function π evaluated when π₯ is equal to four. We notice that our function π of π₯ is an exponential function. So we need to recall the standard rules for differentiating these. We recall that the derivative with respect to π₯ of π to the power of some function π of π₯ is equal to π prime of π₯ multiplied by π to the power of π of π₯. This is a direct application of the chain rule.

Here, our function π of π₯ is six π₯. And applying the power rule of differentiation, we find that π prime of π₯ 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 π prime of π₯ is equal to eight multiplied by π prime of π₯ β thatβs six β multiplied by π to the power of π of π₯ β thatβs π to the power of six π₯. Now, we could simplify and write this as 48π to the power of six π₯ before we substitute π₯ equals four to evaluate its derivative.

But one thing thatβ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 π of π₯ was eight π to the power of six π₯. So if we wanted, we could express π prime of π₯ as six multiplied by π of π₯. When we find π prime of four then, we can give our answer in terms of the original function. In this case, substituting π₯ equals four gives that π prime of four is equal to six multiplied by π of four.

Now, whether you answer the question this way or whether you substitute π₯ equals four into the explicit expression for π prime of π₯, weβll depend on the requirements of the particular question weβre answering. In this case, weβll give our answer as π prime of four is equal to six π of four.