y = f(x) = √ x the line tangent to f(x) is at the point (4, 2). Find the equation of the tangent line.

Question

According to the answer key, the equation is 1/4x+1  but I don't understand where the 1/4 is coming from. I thought the antiderivative of rootx would be (x^(3/2))/(3/2) which is ultimately x^(1/2). But that doesn't seem to be the right route...

Answer 1

The tangent line is finding the derivative, not find the anti-derivative.

 

 

Rewrite the equation using rational exponents.

 

y = x^1/2

 

 

The equation of the tangent line is

 

y = (dy/dx)(x - x₁) + y₁

 

where:

dy/dx is the slope of the tangent line at the target point

(x₁ , y₁) is the point that satisfies the function

 

 

Now we find the slope of the tangent line.  This is the derivative.

 

dy/dx = (1/2)x^-1/2

dy/dx = 1 / (2√x)

dy/dx = 1 / (2√4)

dy/dx = 1 / 4

 

 

Tangent line is then

 

y = (1/4)(x - 4) + 2

 

y = (1/4)x + 1