Which of the following corresponds to the derivative of f(x)=1/x+2 at x = 2, using the alternate definition of a derivative, reduced to its simplest form before taking the limit?
We can use the alternate definition of a derivative to find the derivative of f(x) = 1/x + 2 at x = 2.
The alternate definition of a derivative is given by:
f'(x) = lim(h→0) [f(x + h) - f(x)] / h
Plugging in the given function f(x) = 1/x + 2, we can calculate the derivative:
f'(x) = lim(h→0) [f(x + h) - f(x)] / h
f'(x) = lim(h→0) [(1/(x+h) + 2) - (1/x + 2)] / h
f'(x) = lim(h→0) [(1/(x+h) - 1/x)] / h
To simplify this expression, we need to find a common denominator for the two fractions. Multiplying the first fraction by (x/x) and the second fraction by ((x+h)/(x+h)), we get:
f'(x) = lim(h→0) [(x - (x+h)) / (x(x+h))] / h
f'(x) = lim(h→0) [(x - x - h) / (x(x+h))] / h
f'(x) = lim(h→0) [-h / (x(x+h))] / h
f'(x) = lim(h→0) -1 / (x(x+h))
Taking the limit as h approaches 0, we have:
f'(x) = -1 / (x(x+0))
f'(x) = -1 / (x^2)
Now, we need to find the derivative at x = 2:
f'(2) = -1 / (2^2)
f'(2) = -1 / 4
Therefore, the derivative of f(x) = 1/x + 2 at x = 2, using the alternate definition of a derivative and reduced to its simplest form before taking the limit, is -1/4.
To find the derivative of the function f(x)=1/x+2 using the alternate definition of a derivative, we need to follow these steps:
Step 1: Write the alternate definition of the derivative. The alternate definition of the derivative is given by:
f'(x) = lim(h->0) [(f(x + h) - f(x))/h]
Step 2: Substitute the function f(x) with its expression. In this case, we have:
f(x) = 1/x + 2
Step 3: Substitute x with the given value x = 2. So, we have:
f(2) = 1/2 + 2
Step 4: Simplify the expression.
f(2) = 1/2 + 2 = 1/2 + 4/2 = 5/2
Step 5: Substitute f(x + h) with its expression. We now have:
f(x + h) = 1/(x + h) + 2
Step 6: Substitute the expression for f(x + h) and f(x) into the alternate definition of the derivative.
f'(x) = lim(h->0) [(1/(x + h) + 2 - 1/x - 2)/h]
Step 7: Simplify the expression within the limit before taking the limit.
f'(x) = lim(h->0) [(1/(x + h) - 1/x)/h]
Step 8: Combine the two fractions by obtaining a common denominator.
f'(x) = lim(h->0) [((1*x - 1(x + h))/(x(x + h)))/h]
f'(x) = lim(h->0) [(-h)/(x(x + h))]/h
Step 9: Cancel out h from the numerator and denominator.
f'(x) = lim(h->0) (-1)/(x(x + h))
Step 10: Now, substitute x with the value x = 2 in the expression.
f'(2) = lim(h->0) (-1)/(2(2 + h))
Step 11: Simplify the expression.
f'(2) = lim(h->0) (-1)/(2(2 + h)) = -1/(2(2)) = -1/4
Therefore, the derivative of f(x)=1/x+2 at x = 2, using the alternate definition, reduced to its simplest form before taking the limit, is -1/4.
To find the derivative of a function using the alternate definition, you can use the formula:
f'(x) = lim Δx→0 [f(x + Δx) - f(x)] / Δx
Let's use this formula to find the derivative of f(x) = 1/x + 2 at x = 2.
First, substitute the values into the formula:
f'(2) = lim Δx→0 [f(2 + Δx) - f(2)] / Δx
Now, let's calculate f(2 + Δx):
f(2 + Δx) = 1/(2 + Δx) + 2
Next, substitute the values back into the formula:
f'(2) = lim Δx→0 [1/(2 + Δx) + 2 - (1/2 + 2)] / Δx
Now, simplify the expression:
f'(2) = lim Δx→0 [(1/(2 + Δx) - 1/2)] / Δx
To simplify this expression further, we need to find a common denominator:
f'(2) = lim Δx→0 [(1*2 - (2 + Δx))/(2(2 + Δx))] / Δx
Simplify the numerator:
f'(2) = lim Δx→0 [2 - 2 - Δx] / [2(2 + Δx)Δx]
Simplify the expression:
f'(2) = lim Δx→0 [-Δx] / [2(2 + Δx)Δx]
Cancel out Δx in the numerator and denominator:
f'(2) = lim Δx→0 -1 / [2(2 + Δx)]
Now, take the limit as Δx approaches 0:
f'(2) = -1 / [2(2 + 0)]
f'(2) = -1 / (2 * 2)
f'(2) = -1 / 4
Therefore, the derivative of f(x) = 1/x + 2 at x = 2 using the alternate definition, reduced to its simplest form before taking the limit, is -1/4.