Compute the difference quotient: [(f(x)- f(a))/(x-a)]..
for f(x)= (-1/x^2)
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im sorry, i wrote the question wrong, its supposed to be: f(x)=(-3/x^2), but i think i can follow this same format to solve this one too right?
To compute the difference quotient for the function f(x) = -1/x^2, we can use the definition of the difference quotient and follow these steps:
Step 1: Start with the given function f(x) = -1/x^2.
Step 2: We need to find the value of f(x) and f(a) to substitute them into the difference quotient formula. For this, plug in the values of x and a into the function f(x):
f(x) = -1/x^2
f(a) = -1/a^2
Step 3: Substitute the values of f(x) and f(a) into the difference quotient formula:
[(f(x) - f(a))/(x - a)] = [(-1/x^2) - (-1/a^2)] / (x - a)
Step 4: Simplify the numerator:
[-1/x^2 + 1/a^2] / (x - a)
Step 5: Find a common denominator for the numerator:
[(-a^2 + x^2)/(x^2 * a^2)] / (x - a)
Step 6: Invert the fraction in the denominator and multiply:
[(-a^2 + x^2)/(x^2 * a^2)] * (1/(x - a))
Step 7: Simplify the numerator:
[-(a^2 - x^2)/(x^2 * a^2)] * (1/(x - a))
Step 8: Factor the numerator:
[(x - a)(x + a)/(x^2 * a^2)] * (1/(x - a))
Step 9: Cancel out the common factor of (x - a):
(x + a)/(x^2 * a^2)
So, the difference quotient for the given function f(x) = -1/x^2 is (x + a)/(x^2 * a^2).