### Video Transcript

Evaluate ๐ prime three, where ๐ ๐ฅ is equal to 17๐ฅ squared plus 52๐ฅ.

Well, to solve this problem, we first will need to know what ๐ prime three means. And actually, what it means is the first differential of our function, where the value of ๐ฅ is equal to three. So in order to actually find the value of this, what we need to do is differentiate our function.

And I am gonna do that term by term. And we could do that using this formula where we say that if ๐ฆ is equal to ๐ฅ to the power of ๐, then ๐๐ฆ ๐๐ฅ โ so thatโs the same as ๐ prime cause itโs the first differential of our function, or also the slope function itโs also called โ is equal to ๐ multiplied by ๐ฅ to the power of ๐ minus one.

So what this means in practical terms is we multiply by the exponent. And then we actually reduce the exponent itself by one. Okay, letโs get on and use this to differentiate our function. So if we have the function 17๐ฅ squared plus 52๐ฅ, then when we differentiate this, weโre gonna get the first term equal to 34๐ฅ. And we get that because we actually multiply the coefficient of the ๐ฅ squared by the exponent, so two multiplied by 17. That gives our 34. And then we have ๐ฅ squared. And then what we actually do with this is we take away one from the exponent, so two minus one, which gives us ๐ฅ. So therefore, we get 34๐ฅ.

Now we move on to the second term. And we just get plus 52. And we get 52 because, again, if we multiply the exponent by the coefficient, we get one multiplied by 52, which gives us 52. And then itโs ๐ฅ to the power of one minus one, so ๐ฅ to the power of zero. And we actually have an exponent rule that says that anything to the power of zero is always equal to one. So therefore, that would be 52 multiplied by one, which just gives us 52.

Okay, great! So weโve now differentiated. What we need to do is substitute the value of three in for ๐ฅ, so we can find the value of ๐ prime three. So we get ๐ prime three is equal to 34 multiplied by three. And thatโs because weโve actually substituted in three for ๐ฅ. And then this is plus 52, which is gonna give us 102 plus 52, which is gonna give us 154.

Therefore, we can say that if we have the function ๐ ๐ฅ is equal to 17๐ฅ squared plus 52๐ฅ and then we actually differentiate this to find the slope function and then we substitute ๐ฅ equals three, then weโre gonna get an answer of 154. So what this actually means in practical terms is that, at the point three, if we actually drew the graph of this function, the slope would be equal to 154. And thatโs because we found ๐๐ฆ ๐๐ฅ. And thatโs actually the slope function.