Using the Alternative Formula for Derivatives

Question

Use the formula

f'(x) = lim (z → x) (f(z) − f(x)) / (z − x)

to find the derivative of the functions in Exercises 23–26.

f(x) = x² − 3x + 4

Accepted Answer

Welcome back, everyone. Use the following alternative formula for derivatives to find the derivative of F of X equals X2 minus 2 X + 6. F X equals limit as Z approaches X F C minus F of X divided by c minus X where given four answer choices A says F of X equals 2 x minus 2, B 2 x + 2 C X minus 2, and D X + 2. So let's apply the given definition. We know that F of X is equal to the limit as Z approaches X. Of F of. So what we want to do is simply understand that F of X is X2 minus 2 x + 6. So F of Z basically means that we're replacing all of the Xs with a C. So we get C2 minus 2 Z + 6, and we are subtracting F of X, which is X2 minus 2X + 6, and we have to divide by Z minus X. From here let's simplify. We're going to get the limit as Z approaches X of Z2 minus 2 z + 6 minus X2 + 2 x minus 6. So we're basically distributing the negative sign, and then we have to divide it by c minus X. What we notice is that 6 and -6 can be canceled out. And then we end up with limit as Z approaches X of Z2 minus X2 + 2X minus 2 Z. Divided by C minus X From here we can perform factorization for the difference of squares. We get limit as Z approaches X of Z minus x multiplied by Z + X, right, because we have a difference of 2 squares and then for the other two terms we can take out -2, right? So we can say minus. 2, which leaves us with a C minus 1, Z minus X, I'm sorry. So Z minus X. All of that is divided by Z minus X. And what we can do is simply perform an additional step of factorization. We get limit as C approaches X. We notice that the common factor is C minus X, which leaves us with C + X minus 2 divided by Z minus X, which essentially cancels out the whole. We can cancel out the common factor which is Z minus X, right? And now, We can perform direct substitution. We get Z plus X minus 2 as Z approaches X. So we are replacing our C with X, which gives us X plus X minus 2. And this is simply equal to 2 x minus 2. Which corresponds to the answer choice A. Thank you for watching.

Using the Alternative Formula for Derivatives

Use the formula
f'(x) = lim (z → x) (f(z) − f(x)) / (z − x)
to find the derivative of the functions in Exercises 23–26.

f(x) = x² − 3x + 4

Key Concepts

Derivative

Limit

Alternative Formula for Derivatives

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