Find the derivation of each of the following expressions. a. 3x 2+2x+3 / 2x2 b. (3x+2)(x−2) c. Evaluate Limit…

Question

Find the derivation of each of the following expressions. a. 3x 2+2x+3 / 2x2 b. (3x+2)(x−2) c. Evaluate Limit by factoring Lim x->4 (x2-x-12)/(x2+x−20)

ScanSolve AI Tutor · Accepted Answer

Here's the derivation for question 1.a:

Rewrite the expression by dividing each term in the numerator by the denominator.
$Y = \frac{3x^2 + 2x + 3}{2x^2}$ $Y = \frac{3x^2}{2x^2} + \frac{2x}{2x^2} + \frac{3}{2x^2}$ $Y = \frac{3}{2} + \frac{1}{x} + \frac{3}{2}x^{-2}$ $Y = \frac{3}{2} + x^{-1} + \frac{3}{2}x^{-2}$
Differentiate each term with respect to $x$ .
Recall that $\frac{d}{dx}(c) = 0$ for a constant $c$ , and $\frac{d}{dx}(x^n) = nx^{n-1}$ .
$\frac{dY}{dx} = \frac{d}{dx}\left(\frac{3}{2}\right) + \frac{d}{dx}(x^{-1}) + \frac{d}{dx}\left(\frac{3}{2}x^{-2}\right)$ $\frac{dY}{dx} = 0 + (-1)x^{-1-1} + \frac{3}{2}(-2)x^{-2-1}$ $\frac{dY}{dx} = -x^{-2} - 3x^{-3}$
Rewrite the expression with positive exponents.
$\frac{dY}{dx} = -\frac{1}{x^2} - \frac{3}{x^3}$
The derivation is:
$\boxed{-\frac{1}{x^2} - \frac{3}{x^3}}$

Find the derivation of each of the following expressions. a. 3x 2+2x+3 / 2x2 b. (3x+2)(x-2) c. Evaluate Limit by factoring Lim x->4 (x2-x-12)/(x2+x-20)

Step-by-step solution

Rewrite the expression by dividing each term in the numerator by the denominator.

Differentiate each term with respect to $x$ .

Rewrite the expression with positive exponents.

Still stuck on this one?

More Mathematics Questions

Find the derivation of each of the following expressions. a. 3x 2+2x+3 / 2x2 b. (3x+2)(x-2) c. Evaluate Limit by factoring Lim x->4 (x2-x-12)/(x2+x-20)

Step-by-step solution

Rewrite the expression by dividing each term in the numerator by the denominator.

Differentiate each term with respect to xxx.

Rewrite the expression with positive exponents.

Still stuck on this one?

More Mathematics Questions

Differentiate each term with respect to $x$ .