Question

Find from first principles the derivatives of the following functions:

a) 2x + 5
b) 5x^2
c) x^2 - 2x
d) x(x + 1)

Answer 1

The derivatives of the given functions from first principles are 2, 10x, 2x-2, and 2x+1, respectively, calculated using the limit definition of the derivative which involves taking the limit of the difference quotient as the increment approaches zero.

Step-by-step explanation:

To find the derivatives of the functions from first principles, we use the definition of the derivative which is the limit of the difference quotient as the change in the independent variable approaches zero:

1. For the function f(x) = 2x + 5, the derivative f'(x) is given by:
2. lim h → 0 [(2(x+h) + 5) - (2x + 5)] / h = 2.
3. For the function g(x) = 5x2, the derivative g'(x) is:
4. lim h → 0 [(5(x+h)2) - (5x2)] / h = 10x.
5. For the function h(x) = x2 - 2x, the derivative h'(x) is:
6. lim h → 0 [(x+h)2 - 2(x+h) - (x2 - 2x)] / h = 2x - 2.
7. For the product of functions i(x) = x(x+1), the derivative i'(x) is:
8. lim h → 0 [x+h)(x+h+1) - x(x+1)] / h = 2x + 1.

In this process, we simplify the function inside the limit by expanding, combining like terms, and then dividing by h before taking the limit as h approaches zero. This results in the derivative which represents the slope of the tangent line to the function at any point x.