Using the derivative definition of f ‘ (x) = (f (x+h) – f (x)) / (h) find the derivatives of the following…
1. f(x) = (2x-1) / (x+2)
2. f(x) = sqrt (3x-1)
3. f(x) = 3x^4
my answers were:
1. –x+3
2. (3) / [2(sqrt(3x-1))]
3. 12x^3
are my answers right?
for the first one, I got 5/(x+2)^2
the other two are correct
To confirm if your answers are correct, let's go through each problem step by step using the derivative definition:
1. f(x) = (2x-1) / (x+2)
To find the derivative, plug the function f(x+h) - f(x) / h into the definition:
f'(x) = lim(h->0) {[(2(x+h)-1) / (x+h+2)] - [(2x-1) / (x+2)] / h}
Simplifying this expression, you get:
f'(x) = lim(h->0) {[(2x + 2h - 1) - (2x - 1)(x + h + 2)] / [(x + h + 2)(x + 2)] / h}
Expanding and canceling terms, you should get:
f'(x) = lim(h->0) {[-xh - 3h] / [(x + h + 2)(x + 2)]}
Now, apply the limit as h approaches 0:
f'(x) = (-x) / [(x + 2)^2]
So, your answer for the derivative of f(x) = (2x-1) / (x+2) should be f'(x) = (-x) / [(x + 2)^2]. It seems you made a sign error in your answer, where it should be negative.
2. f(x) = sqrt(3x-1)
Using the derivative definition:
f'(x) = lim(h->0) {[(√(3(x+h)-1) - √(3x-1)] / h}
To simplify, you need to rationalize the numerator:
f'(x) = lim(h->0) {[(√(3(x+h)-1) - √(3x-1)] / h} * [(√(3(x+h)-1) + √(3x-1)] / [(√(3(x+h)-1) + √(3x-1)]
After simplification, you should get:
f'(x) = lim(h->0) {3 / [2(√(3x-1))]}
So, your answer for the derivative of f(x) = sqrt(3x-1) should be f'(x) = 3 / [2(√(3x-1))]. It appears your answer is correct.
3. f(x) = 3x^4
Using the derivative definition:
f'(x) = lim(h->0) {[(3(x+h)^4 - 3x^4)] / h}
Expanding and simplifying, you should get:
f'(x) = lim(h->0) {12x^3 + 12x^2h + 6xh^2 + h^3}
Now, apply the limit as h approaches 0:
f'(x) = 12x^3
Therefore, your answer for f(x) = 3x^4 is correct, f'(x) = 12x^3.
In summary:
1. f'(x) = (-x) / [(x + 2)^2]
2. f'(x) = 3 / [2(√(3x-1))]
3. f'(x) = 12x^3
So, your answers for #2 and #3 are correct, but there was a sign error in your answer for #1.