If an equation of the tangent line to the curve #y = f(x)# at the point where #a = 2# is #y = 4x-5#, find #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)#.