Find an equation of the tangent line to the curve at the point (7, 2).
i got stuck on this one.
Take the derivative.
Find the slope by filling in x in the derivative for the coordinate given.
Set up the equation y = mx + b. Where m is the slope you just calculated. y is your y coordinate value and x is your x coordinate value.
Simply solve for b.
And rewrite in form of y = mx + b (with m and b filled in.)
There are two ways to do this:
1.) Using the Difference Quotient:
$\displaystyle f'(a) = \lim_{h\to0}\frac{f(a+h) - f(a)}{(a+h) - a}$
$\displaystyle f'(7) = \lim_{h\to0}\frac{f(7 + h) - f(7)}{(7 + h) - 7}$
$\displaystyle f'(7) = \lim_{h\to0}\frac{\frac{(7 + h) - 5}{(7 + h) - 6} - 2}{h}$
$\displaystyle f'(7) = \lim_{h\to0}\frac{\frac{(7 + h) - 5 - (14 + 2h) + 12}{(7 + h) - 6}}{h}$
$\displaystyle f'(7) = \lim_{h\to0}\frac{\frac{-h}{h + 1}}{h}$
$\displaystyle f'(7) = \lim_{h\to0}\frac{-1}{h+1}$
$\displaystyle f'(7) = \frac{-1}{1}$
$\displaystyle f'(7) = -1$
2.) Quotient Rule:
$\displaystyle \frac{dy}{dx}\left[\frac{f}{g}\right] = \frac{f'g - fg'}{g^2}$
$\displaystyle \frac{dy}{dx}\left[\frac{x - 5}{x - 6}\right] = \frac{1*(x - 6) - (x - 5)*1}{(x - 6)^2}$
$\displaystyle \frac{dy}{dx}\left[\frac{x-5}{x-6}\right] = \frac{-1}{(x - 6)^2}$
$\displaystyle f'(7) = \frac{-1}{(7-6)^2}$
$\displaystyle f'(7) = \frac{-1}{1}$
$\displaystyle f'(7) = -1$
There you go.
I started it out in exponent form and applied the product rule. This is why I always use the product rule - it is easier to remember. One rule is always easier than two rules. The quotient rule is just another form of the product rule for people who think division is different than multiplication.
I wasn't using the EXACT problem... That's just weird... I was showing the positive version of the rule... I wasn't quoting the problem at all.
I never once said that division and multiplication are different, but going from $\displaystyle g^{-1}$ to $\displaystyle g^{-2}g'$ requires a step that is not always intuitive. The problem sounds like he's in Calc. I. And adding a rule intuitively is not always the best way for someone to solve a problem.
The Reciprocal rule was used, and it is a version of the Quotient Rule, and that Rule was embedded in your solution through the Product Rule.