Which of the following corresponds to the derivative of f(x)=1x+2 at x = 2, using the alternate definition of a derivative, reduced to its simplest form before taking the limit? (1 point) Responses f′(2)=limx→2[−18x] f ' ( 2 ) = lim x → 2 [ − 1 8 x ] f′(2)=limx→2[−14x2] f ' ( 2 ) = lim x → 2 [ − 1 4 x 2 ] f′(2)=limx→2[−116(x−1)] f ' ( 2 ) = lim x → 2 [ − 1 16 ( x − 1 ) ] f′(2)=limx→2[−14(x+2)] f ' ( 2 ) = lim x → 2 [ − 1 4 ( x + 2 ) ] f′(2)=limx→2[−12(x+6)] f ′ ( 2 ) = lim x → 2 [ − 1 2 ( x + 6 ) ]
The correct answer is:
f′(2)=limx→2[-14(x+2)] f ' ( 2 ) = lim x → 2 [ − 1 4 ( x + 2 ) ]
To find the derivative of f(x) = 1x + 2 using the alternate definition of a derivative, we can use the formula:
f'(x) = lim(h→0) [f(x+h) - f(x)] / h
Let's plug in the given function f(x) = 1x + 2 and x = 2 into the formula:
f'(2) = lim(h→0) [(1(2+h) + 2 - (1(2) + 2)] / h
Simplifying, we get:
f'(2) = lim(h→0) [2 + h + 2 - 2 - 2] / h
f'(2) = lim(h→0) [h] / h
Now, we can simplify further by canceling out the h:
f'(2) = lim(h→0) [1]
So, the derivative of f(x) = 1x + 2 at x = 2, using the alternate definition of a derivative, reduced to its simplest form before taking the limit is:
f'(2) = 1
To find the derivative of f(x) = 1x + 2 at x = 2 using the alternate definition of a derivative, we need to start with the definition of the derivative:
f'(x) = lim(h→0) [(f(x + h) - f(x))/h]
First, we substitute the given function f(x) = 1x + 2:
f'(x) = lim(h→0) [((1(x + h) + 2) - (1x + 2))/h]
Next, we simplify the expression within the limit:
f'(x) = lim(h→0) [(1x + 1h + 2 - 1x - 2)/h]
f'(x) = lim(h→0) [(1x - 1x + 1h)/h]
Now, we simplify the numerator:
f'(x) = lim(h→0) [(1h)/h]
f'(x) = lim(h→0) [1]
Finally, we take the limit as h approaches 0:
f'(x) = 1
Therefore, the derivative of f(x) = 1x + 2 at x = 2 using the alternate definition of a derivative is 1.
None of the given options correspond to the correct derivative, as they all introduce additional terms or calculate the derivative incorrectly.