Question

Use the first principle of differentiation to evaluate the derivatives of:

i. 1 + 5x
ii. f(x) = -3x + 2
A. i. 5; ii. -3
B. i. 1; ii. -3
C. i. 0; ii. -3
D. i. 5; ii. 0

Answer 1

Using the first principle of differentiation, the derivative of 1 + 5x is 5, and the derivative of f(x) = -3x + 2 is -3, resulting in option A being correct.

Step-by-step explanation:

To evaluate the derivatives of the given functions using the first principle of differentiation, we follow the definition of the derivative, which is the limit as Δx approaches 0 of the quotient [(f(x+Δx)-f(x))/Δx].

For i. 1 + 5x:

Let f(x) = 1 + 5x.

The derivative, f'(x), is:

lim Δx→0 [(f(x+Δx)-f(x))/Δx] = lim Δx→0 [(1 + 5(x+Δx) - (1 + 5x))/Δx]

= lim Δx→0 [5Δx/Δx]

= lim Δx→0 [5]

= 5.

For ii. f(x) = -3x + 2:

Let f(x) = -3x + 2.

The derivative, f'(x), is:

lim Δx→0 [(f(x+Δx)-f(x))/Δx] = lim Δx→0 [((-3(x+Δx) + 2) - (-3x + 2))/Δx]

= lim Δx→0 [-3Δx/Δx]

= lim Δx→0 [-3]

= -3.

Therefore, the correct answers are: i. 5; ii. -3, which corresponds to option A.