Video: Differentiating a Combination of Exponential Functions Using the Chain Rule

Find the first derivative of the function 𝑦 = (2𝑒^(4π‘₯) βˆ’ 5𝑒^(βˆ’5π‘₯))⁴.

03:29

Video Transcript

Find the first derivative of the function 𝑦 equals two 𝑒 to the power of four π‘₯ minus five 𝑒 to the power of negative five π‘₯ all to the fourth power.

Here, we have a function of a function, sometimes called a composite function. And we’re looking to find the first derivative. That is, we’re going to differentiate 𝑦 with respect to π‘₯ exactly once. Because we’re dealing with a composite function though, we’re going to need to use the chain rule to do so. The chain rule says that if 𝑦 is some function in 𝑒, and 𝑒 itself is some function in π‘₯, then d𝑦 by dπ‘₯ is equal to d𝑦 by d𝑒 times d𝑒 by dπ‘₯.

So, we’re going to need to define our function 𝑒. Here, we’re going to let 𝑒 be the inside function. That’s two 𝑒 to the power of four π‘₯ minus five 𝑒 to the power of negative five π‘₯. And that means we can rewrite 𝑦 as 𝑦 equals 𝑒 to the fourth power. We’re going to need to work out what d𝑒 by dπ‘₯ is and what d𝑦 by d𝑒 is. We’ll start by finding d𝑦 by d𝑒. That’s the derivative of our function 𝑦 with respect to 𝑒 . And here, we recall the rule for differentiating a function in the form π‘Žπ‘₯ to the power of 𝑛 for nonzero real constants π‘Ž and 𝑛.

That is 𝑛 times π‘Žπ‘₯ to the power of 𝑛 minus one. Essentially, we multiply the entire function by the power and then reduce that power by one. So, d𝑦 by d𝑒 is four times 𝑒 to the power of four minus one. That’s four 𝑒 cubed. But how do we differentiate our function 𝑒 with respect to π‘₯? Well, we could recall the fact that the derivative of the function 𝑒 to the power of π‘₯ is 𝑒 to the power of π‘₯ and then use the chain rule to work out the derivative of 𝑒 the power of four π‘₯.

However, it’s absolutely fine to quote the standard result that the derivative of 𝑒 to the power of π‘Žπ‘₯ is π‘Ž times 𝑒 to the power of π‘Žπ‘₯. We also know that to find the derivative of a constant multiplied by a function, we can multiply that constant by the derivative of the function itself. So, we can differentiate 𝑒 to the power of four π‘₯ to get four 𝑒 to the power of four π‘₯. And then, we can multiply that by two. So, that’s two times four 𝑒 to the power of four π‘₯. Similarly, when we differentiate 𝑒 to the power of negative five π‘₯, we get negative five 𝑒 to the power of negative five π‘₯.

So, d𝑒 by dπ‘₯ is two times four 𝑒 to the power of four π‘₯ minus five times negative five 𝑒 to the power of negative five π‘₯. And we obtain d𝑒 by dπ‘₯ to be equal to eight 𝑒 to the power of four π‘₯ plus 25𝑒 to the power of negative five π‘₯. We can now substitute everything we have into our formula for the chain rule. When we do, we see that d𝑦 by dπ‘₯ is equal to four 𝑒 cubed multiplied by eight 𝑒 to the power of four π‘₯ plus 25𝑒 to the power of negative five π‘₯.

We do have a bit of a problem though. We’re wanting to differentiate 𝑦 with respect to π‘₯. And currently we have an expression for the derivative in terms of 𝑒 and π‘₯. So, we go back to our definition of 𝑒. We said that 𝑒 was equal to two 𝑒 to the power four π‘₯ minus five 𝑒 to the power of negative five π‘₯. So, let’s replace 𝑒 in our expression for the derivative with this function. And it becomes four times two 𝑒 to the power of four π‘₯ minus five times 𝑒 to the power of negative five π‘₯ cubed times eight 𝑒 to the power of four π‘₯ plus 25𝑒 to the power of negative five π‘₯. And so, by using the chain rule and quoting some standard results for the derivative of polynomial terms and exponential functions, we found d𝑦 by dπ‘₯.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.