estimate the derivative of the function. when you are unable to eliminate the h in denominator use h=.001
3^x
To estimate the derivative of the function f(x) = 3^x, we can use the definition of the derivative:
f'(x) = lim(h -> 0) [f(x + h) - f(x)] / h.
However, in this case, we are unable to eliminate the h in the denominator, so we will use a small value for h, such as h = 0.001, to approximate the derivative.
To estimate the derivative at a specific point, let's say x = a, the formula becomes:
f'(a) ≈ [f(a + h) - f(a)] / h, where h = 0.001.
Using this formula, we can calculate the approximate derivative of the function at any given point.
For example, let's estimate the derivative at x = 2:
f'(2) ≈ [f(2 + 0.001) - f(2)] / 0.001.
Step 1: Calculate f(2 + 0.001):
f(2 + 0.001) = 3^(2 + 0.001) = 3^2.001.
Step 2: Calculate f(2):
f(2) = 3^2.
Step 3: Substitute the values into the formula:
f'(2) ≈ [3^2.001 - 3^2] / 0.001.
Calculate the numerator using an appropriate calculator:
Numerator ≈ 13.3224 - 9 = 4.3224.
Step 4: Divide the numerator by h:
f'(2) ≈ 4.3224 / 0.001.
Calculate the division:
f'(2) ≈ 4322.4.
Therefore, the estimated derivative of f(x) = 3^x at x = 2 is approximately 4322.4.