Video Transcript
Find the first derivative of the function 𝑦 equals two 𝑒 to the power of four 𝑥 minus five 𝑒 to the power of negative five 𝑥 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 𝑥 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 𝑢, and 𝑢 itself is some function in 𝑥, then d𝑦 by d𝑥 is equal to d𝑦 by d𝑢 times d𝑢 by d𝑥.
So, we’re going to need to define our function 𝑢. Here, we’re going to let 𝑢 be the inside function. That’s two 𝑒 to the power of four 𝑥 minus five 𝑒 to the power of negative five 𝑥. And that means we can rewrite 𝑦 as 𝑦 equals 𝑢 to the fourth power. We’re going to need to work out what d𝑢 by d𝑥 is and what d𝑦 by d𝑢 is. We’ll start by finding d𝑦 by d𝑢. That’s the derivative of our function 𝑦 with respect to 𝑢 . And here, we recall the rule for differentiating a function in the form 𝑎𝑥 to the power of 𝑛 for nonzero real constants 𝑎 and 𝑛.
That is 𝑛 times 𝑎𝑥 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 d𝑢 is four times 𝑢 to the power of four minus one. That’s four 𝑢 cubed. But how do we differentiate our function 𝑢 with respect to 𝑥? Well, we could recall the fact that the derivative of the function 𝑒 to the power of 𝑥 is 𝑒 to the power of 𝑥 and then use the chain rule to work out the derivative of 𝑒 the power of four 𝑥.
However, it’s absolutely fine to quote the standard result that the derivative of 𝑒 to the power of 𝑎𝑥 is 𝑎 times 𝑒 to the power of 𝑎𝑥. 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 𝑥 to get four 𝑒 to the power of four 𝑥. And then, we can multiply that by two. So, that’s two times four 𝑒 to the power of four 𝑥. Similarly, when we differentiate 𝑒 to the power of negative five 𝑥, we get negative five 𝑒 to the power of negative five 𝑥.
So, d𝑢 by d𝑥 is two times four 𝑒 to the power of four 𝑥 minus five times negative five 𝑒 to the power of negative five 𝑥. And we obtain d𝑢 by d𝑥 to be equal to eight 𝑒 to the power of four 𝑥 plus 25𝑒 to the power of negative five 𝑥. We can now substitute everything we have into our formula for the chain rule. When we do, we see that d𝑦 by d𝑥 is equal to four 𝑢 cubed multiplied by eight 𝑒 to the power of four 𝑥 plus 25𝑒 to the power of negative five 𝑥.
We do have a bit of a problem though. We’re wanting to differentiate 𝑦 with respect to 𝑥. And currently we have an expression for the derivative in terms of 𝑢 and 𝑥. So, we go back to our definition of 𝑢. We said that 𝑢 was equal to two 𝑒 to the power four 𝑥 minus five 𝑒 to the power of negative five 𝑥. So, let’s replace 𝑢 in our expression for the derivative with this function. And it becomes four times two 𝑒 to the power of four 𝑥 minus five times 𝑒 to the power of negative five 𝑥 cubed times eight 𝑒 to the power of four 𝑥 plus 25𝑒 to the power of negative five 𝑥. 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 d𝑥.
