Solve 3

**Hallah_az** · March 27th 2006, 06:08 PM

Houw would I solve.
Derivative formula (instantaneous rate of change formula for a function) find the derivative.

a) f(x)=x^2-9x-10

b) Check the answer by then finding the derivative of the same
function above using the short-cut rules for finding derivatives.

**topsquark** · March 28th 2006, 03:47 AM

Originally Posted by Hallah_az

Houw would I solve.
Derivative formula (instantaneous rate of change formula for a function) find the derivative.

a) f(x)=x^2-9x-10

b) Check the answer by then finding the derivative of the same
function above using the short-cut rules for finding derivatives.

a) The general formula (at least for Calc 1) for a derivative is:
$f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$

In this case
$f(x+h)=(x+h)^2-9(x+h)-10$
$=x^2+2hx+h^2-9x-9h-10$
So:
$f(x+h)-f(x)=2hx+h^2-9h$

Thus
$f'(x)=\lim_{h \to 0}=\frac{2hx+h^2-9h}{h}=\lim_{h \to 0}(2x-9+h)=2x-9$

b) The power rule states that the derivative of $ax^n$ is $nax^{n-1}$ and we know the derivative of a constant is zero, so the derivative of $x^2-9x-10$ is:
$2x^{2-1}-9x^{1-1}+0=2x^1-9x^0=2x-9$ .

-Dan

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