# Math Help - Basic Derivative Problem

1. ## Basic Derivative Problem

Greeting MHF, first timer here.

I've been taking Calculus over the summer and I've stumbled upon a question that I need assistance with.. Your help is greatly appreciated

DIRECTIONS: Find the equation of the tangent line to the given function at the given point using: limit h->0 (f(x+h)-f(x))/h

Problem: f(x)= sqrt(x-1), (5,2)

What I've done to try and solve this:

Limit h->0 (sqrt(x+h-1) - sqrt (x-1))/h

Then I tried to multiply the numerators conjugate to both the numerator and the denominator so it looks like now (or at least to me):

limit h->0 (x+h-1-x-1)/h(sqrt(x+h-1)+sqrt(x-1))

Then It simplifies to:

limit h->0 (h-2)/h/h(sqrt(x+h-1)+sqrt(x-1))

but when you use direct substitution to find the slope you get -2/0, which isnt possible..

Where have I gone wrong? Any help is greatly appreciated

2. ## Re: Basic Derivative Problem

$\lim_{h \to 0} \frac{\sqrt{(x+h)-1} - \sqrt{x-1}}{h}$

$\lim_{h \to 0} \frac{\sqrt{(x+h)-1} - \sqrt{x-1}}{h} \cdot \frac{\sqrt{(x+h)-1} + \sqrt{x-1}}{\sqrt{(x+h)-1} + \sqrt{x-1}}$

$\lim_{h \to 0} \frac{(x+h)-1 - (x-1)}{h \left[\sqrt{(x+h)-1} + \sqrt{x-1}\right]}$

$\lim_{h \to 0} \frac{h}{h \left[\sqrt{(x+h)-1} + \sqrt{x-1}\right]}$

now finish it ...

3. ## Re: Basic Derivative Problem

From that last equation you can cancel out the h's on the outside so you get

lim h->0 1/(sqrt(x-h+1)+sqrt(x-1)

Direct substitution and you get m=1/4

Thanks. I was able to get the equation using point slope.