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QuestionFind the difference quotient f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h} for f(x)=3x2f(x)=\frac{3}{x^{2}}, where h≠0h \neq 0.

Studdy Solution

STEP 1

Assumptions1. The function is f(x)=3xf(x)=\frac{3}{x^{}} . We are looking for the difference quotient, which is defined as f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}, where h≠0h \neq0

STEP 2

We first need to find f(x+h)f(x+h) by substituting x+hx+h into the function f(x)f(x).
f(x+h)=(x+h)2f(x+h) = \frac{}{(x+h)^{2}}

STEP 3

Next, we need to find f(x+h)−f(x)f(x+h)-f(x) by subtracting f(x)f(x) from f(x+h)f(x+h).
f(x+h)−f(x)=3(x+h)2−3x2f(x+h)-f(x) = \frac{3}{(x+h)^{2}} - \frac{3}{x^{2}}

STEP 4

To simplify the expression, we can find a common denominator for the two fractions.
f(x+h)−f(x)=3x2−3(x+h)2x2(x+h)2f(x+h)-f(x) = \frac{3x^{2} -3(x+h)^{2}}{x^{2}(x+h)^{2}}

STEP 5

Next, we need to expand the numerator.
f(x+h)−f(x)=3x2−3(x2+2xh+h2)x2(x+h)2f(x+h)-f(x) = \frac{3x^{2} -3(x^{2}+2xh+h^{2})}{x^{2}(x+h)^{2}}

STEP 6

implify the numerator.
f(x+h)−f(x)=−3h2−6xhx2(x+h)2f(x+h)-f(x) = \frac{-3h^{2}-6xh}{x^{2}(x+h)^{2}}

STEP 7

Now, we can find the difference quotient by dividing f(x+h)−f(x)f(x+h)-f(x) by hh.
f(x+h)−f(x)h=−3h2−6xhx2(x+h)2h\frac{f(x+h)-f(x)}{h} = \frac{\frac{-3h^{2}-6xh}{x^{2}(x+h)^{2}}}{h}

STEP 8

implify the expression by cancelling out the hh in the numerator and the denominator.
f(x+h)−f(x)h=−3h−6xx2(x+h)2\frac{f(x+h)-f(x)}{h} = \frac{-3h-6x}{x^{2}(x+h)^{2}}The difference quotient for f(x)=3x2f(x)=\frac{3}{x^{2}} is −3h−6xx2(x+h)2\frac{-3h-6x}{x^{2}(x+h)^{2}}.

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