Differential the Following by First Principle: a. Y = 2x2 b. Y = 1/X. Evaluate The Following: Lim (2x2 + 2x…

Question

Differential the Following by First Principle: a. Y = 2x2 b. Y = 1/X. Evaluate The Following: Lim (2x2 + 2x − 1) X→2

ScanSolve AI Tutor · Accepted Answer

here's the solution for question 2.a, differentiating by the first principle:

The first principle of differentiation states that for a function $Y = f(x)$ , its derivative $\frac{dY}{dx}$ is given by:

\frac{dY}{dx} = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

Identify $f(x)$ and $f(x+h)$ .
Given $Y = f(x) = 2x^2$ .
Then, $f(x+h) = 2(x+h)^2$ .
Expand $f(x+h)$ .
$f(x+h) = 2(x^2 + 2xh + h^2)$ $f(x+h) = 2x^2 + 4xh + 2h^2$
Calculate $f(x+h) - f(x)$ .
$f(x+h) - f(x) = (2x^2 + 4xh + 2h^2) - (2x^2)$ $f(x+h) - f(x) = 4xh + 2h^2$
Divide by $h$ .
$\frac{f(x+h) - f(x)}{h} = \frac{4xh + 2h^2}{h}$ $\frac{f(x+h) - f(x)}{h} = \frac{h(4x + 2h)}{h}$ $\frac{f(x+h) - f(x)}{h} = 4x + 2h$
Take the limit as $h \to 0$ .
$\frac{dY}{dx} = \lim_{h \to 0} (4x + 2h)$ $\frac{dY}{dx} = 4x + 2(0)$ $\frac{dY}{dx} = 4x$
The derivative by first principle is:
$\boxed{4x}$

Differential the Following by First Principle: a. Y = 2x2 b. Y = 1/X. Evaluate The Following: Lim (2x2 + 2x - 1) X→2