This limit represents the derivative of some function f at some number a. Select an appropriate f and a.
1f(x) = x4, a = 4
f(x) = x1/4, a = 4
f(x) = x1/4, a = 256
f(x) = �ãx, a = 256
f(x) = �ãx, a = 16
To determine which function and value of a represent the derivative of some function f, we need to find the limit that represents the derivative.
The derivative of a function f at a number a can be represented as:
f'(a) = lim(h->0) [(f(a+h) - f(a)) / h]
For f(x) = x^4 and a = 4:
f'(4) = lim(h->0) [(f(4+h) - f(4)) / h]
= lim(h->0) [((4+h)^4 - 4^4) / h]
= lim(h->0) [(256 + 64h + 6h^2 + h^3) / h]
= lim(h->0) [256/h + 64 + 6h + h^2]
Evaluating this limit will give us the derivative of f at a.
Similarly, we can go through the same process for the other functions and values of a to determine the correct representation of the derivative.
For f(x) = x^(1/4) and a = 4:
f'(4) = lim(h->0) [(f(4+h) - f(4)) / h]
= lim(h->0) [((4+h)^(1/4) - 4^(1/4)) / h]
For f(x) = x^(1/4) and a = 256:
f'(256) = lim(h->0) [(f(256+h) - f(256)) / h]
= lim(h->0) [((256+h)^(1/4) - 256^(1/4)) / h]
For f(x) = √x and a = 256:
f'(256) = lim(h->0) [(f(256+h) - f(256)) / h]
= lim(h->0) [((256+h)^(1/2) - 256^(1/2)) / h]
For f(x) = √x and a = 16:
f'(16) = lim(h->0) [(f(16+h) - f(16)) / h]
= lim(h->0) [((16+h)^(1/2) - 16^(1/2)) / h]
Now, we can evaluate these limits to find the derivatives of each function at their respective values of a.