Video: Finding the First Derivative of an Exponential Function

Find d𝑦/dπ‘₯ if 5𝑦𝑒^(2π‘₯) = 7𝑒⁡.

01:48

Video Transcript

Find d𝑦 by dπ‘₯ if five 𝑦𝑒 to the two π‘₯ 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 π‘₯. 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 π‘₯. Now, actually, one over 𝑒 to the power of two π‘₯ is equal to 𝑒 to the negative two π‘₯. 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 π‘₯.

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 π‘˜π‘₯ with respect to π‘₯ is π‘˜π‘’ to the π‘˜π‘₯. And of course, remembering that the constant factor rule allows us to take constants outside a derivative and concentrate on differentiating the function of π‘₯ itself.

In other words, d𝑦 by dπ‘₯ is equal to seven 𝑒 to the power of five over five times the derivative of 𝑒 to the negative two π‘₯ with respect to π‘₯. And the derivative of 𝑒 to the negative two π‘₯ with respect to π‘₯ is negative two 𝑒 to the negative two π‘₯. This means d𝑦 by dπ‘₯ is negative two times seven 𝑒 to the power of five over five times 𝑒 to the negative two π‘₯.

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 π‘₯. And we can therefore say that d𝑦 by dπ‘₯ is equal to negative two 𝑦.

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