Video: Finding the First Derivative of an Exponential Function

Video Transcript

Find d𝑦 by d𝑥 if five 𝑦𝑒 to the two 𝑥 equals seven 𝑒 to the five.

Now, at first glance, this does look a little complicated. However, we can clearly see that we can rearrange the equation to make 𝑦 the subject. We’re going to divide both sides of the equation by five 𝑒 to the two 𝑥. On the left-hand side, that leaves us simply with 𝑦. And on the right-hand side, we have seven 𝑒 to the power of five over five 𝑒 to the two 𝑥. Now, actually, one over 𝑒 to the power of two 𝑥 is equal to 𝑒 to the negative two 𝑥. So we can rewrite our equation. And we say that 𝑦 is equal to seven 𝑒 to the power of five over five times 𝑒 to the negative two 𝑥.

Notice that seven 𝑒 to the five over five is just a constant. So we can differentiate this using the general formula for the derivative of the exponential function. The derivative of 𝑒 to the 𝑘𝑥 with respect to 𝑥 is 𝑘𝑒 to the 𝑘𝑥. And of course, remembering that the constant factor rule allows us to take constants outside a derivative and concentrate on differentiating the function of 𝑥 itself.

In other words, d𝑦 by d𝑥 is equal to seven 𝑒 to the power of five over five times the derivative of 𝑒 to the negative two 𝑥 with respect to 𝑥. And the derivative of 𝑒 to the negative two 𝑥 with respect to 𝑥 is negative two 𝑒 to the negative two 𝑥. This means d𝑦 by d𝑥 is negative two times seven 𝑒 to the power of five over five times 𝑒 to the negative two 𝑥.

Notice that the derivative can actually be expressed in terms of 𝑦 since we said that 𝑦 was equal to seven 𝑒 to the power of five over five times 𝑒 to the negative two 𝑥. And we can therefore say that d𝑦 by d𝑥 is equal to negative two 𝑦.