Consider the function g( x ) = √ (x + 1) Use the definition of derivative (limits) to find g ′ ( 3 ) Find the equation of the tangent line to the curve y = g ( x ) at the point x = 3.

Question

Graph the curve of the function and the tangent line to the curve at x = 3 on the same set of axes.

Answer 1

Using the definition of derivative:

g'(x) = lim_h→0 [ g (x + h) - g (x)] / h

So, replacing g(x) = √(x+1) we got

g'(x) = lim_h→0 [ √(x + h + 1) - √(x+1)] / h I will call this equation (i).

If we multiply (i) by [√(x + h + 1) + √(x+1)] / [√(x + h + 1) + √(x+1)] we will have:

g'(x) = lim_h→0 [√(x + h + 1) - √(x+1)] * [√(x + h + 1) + √(x+1)] / [h*[√(x + h + 1) + √(x+1)]] let's call the numerator as (ii).

Solving (ii):

(x + h + 1) + √(x + h + 1)*(x + 1) - √(x + h + 1)*(x + 1) - (x + 1) = h

Using the result of (ii) in the g'(x) equation:

g'(x) = lim_h→0 h / [h*[√(x + h + 1) + √(x+1)]]

We can simplify and use the fact that h is going to 0, and finally find that:

g'(x) = 1 / [ 2√(x+1)]

if x = 3, g'(3) = 1/4

The tangent line equation is given by:

y = g(a) +g'(a)(x - a)

In this case g(a=3) = √3 +1 = 2

Then;

y(x) = 2 + (x - 3)/4