Let f(x) = x².
a. Show that f(x)−f(y) / x−y = f′(x+y²), ...

Question

Let f(x) = x².

a. Show that f(x)−f(y) / x−y = f′(x+y²), for all x≠y.

Accepted Answer

Welcome back, everyone. Consider the function G of T equals T2. Demonstrate that G T minus GS divided by T minus X equals G T + S divided by 2 for all values of T that aren't equal to S. Where G of T + S divided by 2 is the derivative of G. Of T plus S divided by 2. A says the expressions are equal. B says they are not. C says the relationship holds only for specific values of T and S, while B says it holds only for specific values of T. Now, for us to demonstrate this equality, then we'll need to find GFT minus GFS divided by T minus S and G T + S divided by 2. That's what we need to prove. Now, since we already know that G of T equals T2d, let's first start by computing the difference quotient, OK? So that means we'll be finding G T minus G of S divided by T minus S. And no GS to find GFS we substitute S into our function. So this is gonna be T2 minus Squad divided by T minus S. Now, we can simplify this even further using the difference of 2 squares, OK? Because we're called that by the difference of 2 squares, we can write T2 minus Squared as T minus S multiplied by T + S. And when we do that, then no, we'll be able to cancel T minus S as a factor from the numerator and denominator, and we are left with T plus S for all values of T not being equal to S. Now, let's try the right-hand side of our problem. First, we need to compute the derivative, OK? Now we know that the derivative of GFT is going to be equal to 2 T. OK? And we differentiate T squared. So that tells us then. That G. G T + S divided by 2 is going to be equal to 2 multiplied by T + S divided by 2. And now if we cancel 2 from the numerator and denominator we're left with T + S. Thus this tells us that The difference quotient, the result of the difference quotient T + S is equal to the derivative of T + S divided by 2. Both sides are equal, therefore, A is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Let f(x) = x².
a. Show that f(x)−f(y) / x−y = f′(x+y²), for all x≠y.

Key Concepts

Difference Quotient

Derivative

Function Composition

Watch next