Answer:
Explanation:
The first principles method for differentiation involves finding the limit of the difference quotient as h approaches 0. The difference quotient for a function f(x) is given by:
[f(x + h) - f(x)]/h
a) f(x) = 4x
[f(x + h) - f(x)]/h = [4(x + h) - 4x]/h = (4x + 4h - 4x)/h = 4
Therefore, the derivative of f(x) = 4x is 4.
b) f(x) = -3x
[f(x + h) - f(x)]/h = [-3(x + h) - (-3x)]/h = (-3x - 3h + 3x)/h = -3
Therefore, the derivative of f(x) = -3x is -3.
c) g(x) = x^2 - 4
[f(x + h) - f(x)]/h = [(x + h)^2 - 4 - (x^2 - 4)]/h
= [(x^2 + 2hx + h^2) - x^2]/h = 2x + h
Therefore, the derivative of g(x) = x^2 - 4 is 2x.
d) f(x) = 2x + 5
[f(x + h) - f(x)]/h = [2(x + h) + 5 - (2x + 5)]/h
= (2x + 2h + 5 - 2x - 5)/h = 2
Therefore, the derivative of f(x) = 2x + 5 is 2.