Find the first derivative of 𝑓 of 𝑥 is equal to nine 𝑥 squared minus 𝑥 minus seven multiplied by seven 𝑥 squared minus eight 𝑥 minus seven at negative one, 24.
Well, the first thing we could do when we take a look at our function is think, right, well, to differentiate this, we’re gonna expand the parenthesis and just multiply out. However, to solve it more succinctly, what we can do is use the product rule. So to use the product rule, I’m gonna call our first parenthesis 𝑢 and our second parenthesis 𝑣. And the product rule states that if we’ve got 𝑦 is equal to 𝑢𝑣 — so in this case, our function is equal to 𝑢𝑣 — then d𝑦 d𝑥, so the first derivative, is equal to 𝑢d𝑣 d𝑥 plus 𝑣d𝑢 d𝑥. What this means in practice is 𝑢 multiplied by the derivative of 𝑣 and 𝑣 multiplied by the first derivative of 𝑢.
So the first thing I’m gonna do is I’m gonna take 𝑢 which is nine 𝑥 squared minus 𝑥 minus seven. And what I’m gonna do is I’m going to differentiate this. And when I do this, I’m gonna get 18𝑥 minus one. And the way I get that, just to remind us of how we differentiate, is we multiply the exponents, so two, by the coefficient, which was nine. And then we reduce the exponent by one, so we take two minus one. So we’re gonna get 18𝑥. And then if we differentiate 𝑥, we just get one. And if we differentiate seven or any integer, we get zero.
So now, we’re gonna do the same to 𝑣. So 𝑣 is equal to seven 𝑥 squared minus eight 𝑥 minus seven. And if we differentiate this, again using the same method, we’re gonna get 14𝑥 minus eight. So we got 14𝑥 because we did two multiplied by seven which gives us 14 and then reduce the exponent from two to one. So we just get 14𝑥. And then, as we said, eight 𝑥 just differentiates the eight and the seven will tend to zero. So we got 14𝑥 minus eight. Great, so now, we’ve got all the parts we need to be able to use the product rule.
So, therefore, we can say that d𝑦 d𝑥 is going to be equal to, first of all, 𝑢 d𝑣 d𝑥, so nine 𝑥 squared minus 𝑥 minus seven multiplied by 14𝑥 minus eight. And then added to this, we have 𝑣 d𝑢 d𝑥 which is gonna be seven 𝑥 squared minus eight 𝑥 minus seven multiplied by 18𝑥 minus one. So now, if the question just said, find the first derivative of our function, so I’ve drawn that I just wanted us to find the first derivative, then what I’d do is that I’d simplify this by expanding out the parenthesis and then cancelling or collecting terms where possible.
However, we’re asked to find the first derivative at a specific point, so at a value of 𝑥 that we know. So, therefore, I’m not gonna simplify this any further. I’m going to substitute in the value of 𝑥 and calculate the value of our first derivative. So what I’m gonna do is I’m gonna substitute in 𝑥 equals negative one because that’s our 𝑥-value from our coordinates. So then when we substitute this in, we’re gonna get d𝑦 d𝑥 is equal to nine multiplied by negative one all squared minus negative one minus seven multiplied by 14 multiplied by negative one minus eight plus seven multiplied by negative one all squared minus eight multiplied by negative one minus seven multiplied by 18 multiplied by negative one minus one.
So when simplified, this is gonna give us nine. And that’s because nine multiplied by negative one squared, well, negative one squared is a negative multiplied by a negative which gives us positive, so then plus one minus seven multiplied by negative 14 minus eight plus seven minus negative eight minus seven multiplied by negative 18 minus one. So this is gonna give us three multiplied by negative 22 plus eight multiplied by negative 19.
And in the second parenthesis, we know we had eight at the beginning because we had seven minus negative eight. Well, if we subtract the negatives, it’s the same as add. So seven add eight minus seven just gives us eight. Great, so now we can calculate this to find the first derivative. Well, this gives us the value of negative 218. So, therefore, we can say that the first derivative of 𝑓 of 𝑥 is equal to nine 𝑥 squared minus 𝑥 minus seven multiplied by seven 𝑥 squared minus eight 𝑥 minus seven at the point negative one, 24 is gonna have the value negative 218.