Use the Limit Definition to Find the Derivative g(x)=9-x^2

Problem

g(x)=9−x2

Solution

1. State the limit definition of the derivative for the function g(x)

d(g(x))/d(x)=(lim_h→0)((g*(x+h)−g(x))/h)

2. Substitute the function g(x)=9−x2 into the definition.

d(g(x))/d(x)=(lim_h→0)(((9−(x+h)2)−(9−x2))/h)

3. Expand the squared binomial (x+h)2 inside the numerator.

d(g(x))/d(x)=(lim_h→0)((9−(x2+2*x*h+h2)−9+x2)/h)

4. Distribute the negative sign and simplify the numerator by canceling terms.

d(g(x))/d(x)=(lim_h→0)((9−x2−2*x*h−h2−9+x2)/h)

5. Combine like terms to isolate the terms containing h

d(g(x))/d(x)=(lim_h→0)((−2*x*h−h2)/h)

6. Factor out h from the numerator to prepare for division.

d(g(x))/d(x)=(lim_h→0)((h*(−2*x−h))/h)

7. Divide out the common factor h from the numerator and denominator.

d(g(x))/d(x)=(lim_h→0)(−2*x−h)

8. Evaluate the limit by substituting h=0 into the expression.

d(g(x))/d(x)=−2*x−0

Final Answer

d(g(x))/d(x)=−2*x

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