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If 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) = 3# but how do I find #f'(2)#?

My textbook says #f'(2) = 4# but how exactly?

Answer 1

Aurora Alvarado

# f'(2)=4#

The tangent to the curve #y=f(x)# is

# y=4x-5 #

Comparing with #y=mx+c # we see that #m=4# so the gradient is #4#, and the gradient is also #f'(2)#

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Answer 2

Ezra Adams

To find f'(2), we need to determine the derivative of the function f(x) at x = 2. Since the tangent line to the curve at x = 2 is given as y = 4x - 5, we can use this information to find the derivative.

The equation of the tangent line y = 4x - 5 is in the form y = mx + b, where m represents the slope of the line. In this case, the slope is 4.

The derivative of a function represents the slope of the tangent line at any given point. Therefore, f'(2) is equal to the slope of the tangent line at x = 2, which is 4.

Hence, f'(2) = 4.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.