Evaluate 𝑓 prime of one, where 𝑓 of 𝑥 is equal to one minus six divided by three 𝑥 minus five.
The question gives us a function 𝑓 of 𝑥, and it wants us to find 𝑓 prime of one. That’s the first derivative of 𝑓 evaluated at 𝑥 is equal to one. So let’s start by finding an expression for 𝑓 prime of 𝑥. That’s the derivative of one minus six divided by three 𝑥 minus five with respect to 𝑥.
We can simplify this by using the fact that the derivative of the difference between two functions is equal to the difference of their derivatives. This gives us the derivative of one with respect to 𝑥 minus the derivative of six divided by three 𝑥 minus five with respect to 𝑥. However, one is just a constant, so its derivative is equal zero.
To evaluate our second derivative, we see it’s the quotient to two functions. So we’ll do this by using the quotient rule. The quotient rule tells us for functions 𝑢 and 𝑣, the derivative of 𝑢 divided by 𝑣 is equal to 𝑣𝑢 prime minus 𝑢𝑣 prime all divided by 𝑣 squared. We’ll set 𝑢 to be the function in our numerator. That’s six. And we’ll set 𝑣 equal to the function in our denominator. That’s three 𝑥 minus five.
We now want to find expressions for 𝑢 prime and 𝑣 prime. First, we see that 𝑢 is a constant six. So its derivative is equal to zero. Next, to differentiate three 𝑥 minus five, we recall the power rule for differentiation. To differentiate 𝑎𝑥 to the 𝑛th power, we multiply by the exponent and reduce the exponent by one. This tells us the derivative of three 𝑥 is equal to three. And we know the derivative of the constant negative five is equal to zero.
We’re now ready to find the derivative of this quotient by using the quotient rule. And remember, we need to multiply this value by negative one. Substituting our values for 𝑢, 𝑣, 𝑢 prime, and 𝑣 prime, we get negative one times three 𝑥 minus five times zero minus six times three all divided by three 𝑥 minus five squared. And we can simplify this expression. Three 𝑥 minus five times zero is equal to zero. And then negative one times negative six times three is equal to 18.
So we’ve found an expression for 𝑓 prime of 𝑥. It’s equal to 18 divided by three 𝑥 minus five squared. But remember, the question wants us to find the value of 𝑓 prime of 𝑥 when 𝑥 is equal to one. So we need to substitute 𝑥 is equal to one into our expression for 𝑓 prime of 𝑥. Doing this, we get 𝑓 prime of one is equal to 18 divided by three times one minus five squared.
And finally, we can just evaluate this expression. We have three times one minus five is equal to negative two. So we get 18 divided by negative two squared. And negative two squared is equal to four. Finally, we simplify 18 divided by four to get nine divided by two. Therefore, we’ve shown if 𝑓 of 𝑥 is equal to one minus six divided by three 𝑥 minus five, then 𝑓 prime of 𝑥 evaluated at 𝑥 is equal to one is equal to nine divided by two.