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QuestionFind the difference quotient f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h} for f(x)=9x2f(x)=\frac{9}{x^{2}}, simplifying where h≠0h \neq 0.

Studdy Solution

STEP 1

Assumptions1. The function is f(x)=9xf(x)=\frac{9}{x^{}} . We are asked to find the difference quotient, which is f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}, where h≠0h \neq0

STEP 2

First, we need to find f(x+h)f(x+h). This is done by replacing xx in the function f(x)f(x) with (x+h)(x+h).
f(x+h)=9(x+h)2f(x+h) = \frac{9}{(x+h)^{2}}

STEP 3

Now, we need to find f(x+h)−f(x)f(x+h)-f(x). This is done by subtracting f(x)f(x) from f(x+h)f(x+h).
f(x+h)−f(x)=9(x+h)2−9x2f(x+h)-f(x) = \frac{9}{(x+h)^{2}} - \frac{9}{x^{2}}

STEP 4

To simplify this expression, we need to find a common denominator for the two fractions. The common denominator of (x+h)2(x+h)^{2} and x2x^{2} is x2(x+h)2x^{2}(x+h)^{2}.
f(x+h)−f(x)=9x2−9(x+h)2x2(x+h)2f(x+h)-f(x) = \frac{9x^{2} -9(x+h)^{2}}{x^{2}(x+h)^{2}}

STEP 5

Expand the numerator to simplify further.
f(x+h)−f(x)=9x2−9(x2+2xh+h2)x2(x+h)2f(x+h)-f(x) = \frac{9x^{2} -9(x^{2}+2xh+h^{2})}{x^{2}(x+h)^{2}}

STEP 6

implify the numerator.
f(x+h)−f(x)=−9h(2x+h)x2(x+h)2f(x+h)-f(x) = \frac{-9h(2x+h)}{x^{2}(x+h)^{2}}

STEP 7

Now, we need to find the difference quotient, which is f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}.
f(x+h)−f(x)h=−9h(2x+h)h⋅x2(x+h)2\frac{f(x+h)-f(x)}{h} = \frac{-9h(2x+h)}{h \cdot x^{2}(x+h)^{2}}

STEP 8

implify the expression by canceling out hh from the numerator and denominator.
f(x+h)−f(x)h=−(2x+h)x2(x+h)2\frac{f(x+h)-f(x)}{h} = \frac{-(2x+h)}{x^{2}(x+h)^{2}}The difference quotient for f(x)=x2f(x)=\frac{}{x^{2}} is −(2x+h)x2(x+h)2\frac{-(2x+h)}{x^{2}(x+h)^{2}}.

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