If an equation of the tangent line to the curve $y = f (x)$ $y = f(x)$ at the point where $a = 2$ $a = 2$ is $y = 4 x - 5$ $y = 4x-5$ , find $f (2)$ $f(2)$ and $f'(2)$ ? I know f(2) is 3 but how do I find $f'(2)$ ?

Question

The textbook says that $f'(2) = 4$ but how exactly?

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Answer 1 · 2017-06-24T16:49:33

I presume the statement "where $a=2$ " should read "where $x=2$ "

The gradient of the tangent to a curve at any particular point is given by the derivative of the curve at that point.

We are given that the equation of the tangent when $x=2$ is:

$y=4x-5$

Comparing with the standard form equation of a straight line:

$y=mx+c$

We see that:

$m=4$

And as indicated above this is the same as the value of the derivative when $x=2$ , thus:

$f'(2) = 4$ QED

Without additional information, we cannot infer any information regarding $f(2)$ .