Answer to Question #187523 in Calculus for kris

Question

1. Use the first principal to differentiate the following with respect to x

f(x) = 3x²- 4x + 7

Answer 1

Solution. Using differentiation from first principles

$$\frac{dy}{dx}=\lim\limits_{h\to0}\frac{f(x+h)-f(x)}{h}$$

According to the condition of the problem f(x)=3x²-4x+7. Therefore

$$f(x+h)=3(x+h)^2-4(x+h)+7=3x^2+6hx+3h^2-4x-4h+7.$$

Hence

$$f(x+h)-f(x)=3x^2+6hx+3h^2-4x-4h+7-3x^2+4h-7$$ $$f(x+h)-f(x)=6hx+3h^2-4h$$

As result get

$$\frac{dy}{dx}=\lim\limits_{h\to0}\frac{6hx+3h^2-4h}{h}=\lim\limits_{h\to0}(6x+3h-4)=6x-4$$

Answer.

$$\frac{dy}{dx}=6x-4$$