Suppose that you can calculate the derivative of a function using the formula f′(x)=2f(x)+4x. If the output value of the function at x=4 is 4 estimate the value of the function at 4.006.

Answer 1

f(4+0.006) = f(4) + f'(4) * 0.006

f' is the slope of the tangent line. Thus, since ∆y/∆x ≈ dy/dx,
the tangent line at x=4 is
y-f(4) ≈ f'(4) (x-4)
That is, the equation shown above