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The first-order derivative of a function of the form y=f(x) evaluated at x=2 is: a. The...

The first-order derivative of a function of the form y=f(x) evaluated at x=2 is:

a. The rate of change [Delta_y/Delta_x], where Delta_x=3-2 and Delta_y=f(3)-f(2).

b. The slope of the line that is tangent to y=f(x) at the point (2,f(2)) in the (x,y) Cartesian space.

c. The slope of the line that is tangent to y=f’(x) at the point (2,f’(2)) in the (x,y’) Cartesian space.

d. None of the above.

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Answer #1

Answer - b

The first order derivative measures how much 'y' changes relative to a very small change in x.

The option 'a' is a very crude approximation. It measures how 'y' changes for a unit increase in 'x'. But first derivative looks at change in 'y' for a very very small change in 'x'.

The derivative at any point is equal to the slope of the tangent to the function at that particular point. When we measure the slope of f(x) at (2,f(2)), we are checking how y changes with respect to x at that particular point. Hence this is the correct option.

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