# Video: Differentiating Functions Involving Exponential Functions Using the Product Rule

Chris OβReilly

Find the first derivative of the function 𝑦 = 2𝑥^(8)𝑒^(5𝑥).

03:39

### Video Transcript

Find the first derivative of the function π¦ equals two π₯ to the power of eight π to the power of five π₯.

Well, we can actually look at our function and see thatβs in the form π¦ equals π’π£. And therefore, what we can actually use is the product rule to actually help us differentiate. And the product rule tells us that ππ¦ ππ₯ equals π’ ππ£ ππ₯ plus π£ ππ’ ππ₯. So therefore, what we need to do is decide in our function, if we want to find the first derivative, what π’ and π£ are going to be.

Well, Iβve actually decided that π’ is gonna be equal to two π₯ to the power of eight. And π£ is equal to π to the power of five π₯. So now what we need to do is actually find ππ’ ππ₯ and ππ£ ππ₯. Iβm gonna start with ππ’ ππ₯. And Iβll do that by differentiating two π₯ to the power of eight, which would just give me 16π₯ to the power of seven. And we get that same way that we differentiate any term. Weβve multiplied the exponent by the coefficient, so eight multiplied by two, which gives us 16. And now weβd reduced the exponent by one, so from eight to seven.

Okay, now we move on to π£ to find ππ£ ππ₯. So now to find ππ£ ππ₯, what weβre actually gonna use is a rule. And that rule states that if π¦ is equal to π to the power of π π₯, then ππ¦ ππ₯ is equal to the derivative of π π₯ multiplied by π to the power of π π₯. And we actually get this from the chain rule, because if we have ππ¦ ππ₯ is equal to ππ¦ ππ’ multiplied by ππ’ ππ₯, then we will have π’ is equal to π π₯ and π¦ is equal to π to the power of π’.

So therefore, weβd have that ππ¦ ππ₯ is equal to the derivative of π π₯ multiplied by π to the power of π’. And thatβs because the derivative of π to the power of π’ is just π to the power of π’ because π is a number for which the gradient of π to the power of π₯ is π to the power of π₯. Okay, so then if we substitute back in for π’, we just get the first derivative of π π₯ multiplied by π to the power of π π₯.

Okay, great! So weβve just seen where weβve actually got this rule from. Letβs use it to actually carry on and differentiate our function. Well, weβre gonna get ππ£ ππ₯ is equal to five multiplied by π to the power of five π₯. And thatβs because the derivative of five π₯ is just five. So weβve got the derivative of π π₯, which was five π₯, multiplied by π to the power of five π₯.

Okay, so now at this stage, we can actually use the product rule to actually find out what ππ¦ ππ₯ or the first root of our function is going to be. So first of all, we have π’, which is two π₯ to the power of eight, multiplied by ππ£ ππ₯, which is five π to the power of five π₯. And this is plus π to the power of five π₯, because thatβs our π£ multiplied by our ππ’ ππ₯, which is 16π₯ to the power of seven.

Okay, so we now are just gonna rearrange this to find our ππ¦ ππ₯ and just tidy up. So therefore, weβre gonna get that the first derivative of our function π¦ equals two π₯ to the power of eight π to the power of five π₯ is equal to 10π₯ to the power of eight multiplied by five π to the power of five π₯ plus 16π₯ to the power of seven multiplied by π to the power of five π₯.