Page:Calculus Made Easy.pdf/151

 (7) $$\dfrac{x^2-3x+1}{(x-1)(x+2)(x-3)}$$.

(8) $$\dfrac{5x^2+7x+1}{(2x+1)(3x-2)(3x+1)}$$.

(9) $$\dfrac{x^2}{x^3 - 1}$$.

(10) $$\dfrac{x^4+1}{x^3+1}$$.

(11) $$\dfrac{5x^2+6x+4}{(x+1)(x^2+x+1)}$$.

(12) $$\dfrac{x}{(x-1)(x-2)^2}$$.

(13) $$\dfrac{x}{(x^2-1)(x+1)}$$.

(14) $$\dfrac{x+3}{(x+2)^2(x-1)}$$.

(15) $$\dfrac{3x^2+2x+1}{(x+2)(x^2+x+1)^2}$$.

(16) $$\dfrac{5x^2+8x-12}{(x+4)^3}$$.

(17) $$\dfrac{7x^2+9x-1}{(3x-2)^4}$$.

(18) $$\dfrac{x^2}{(x^3-8)(x-2)}$$.

Consider the function (see p. 14) $$y=3x$$; it can be expressed in the form $$x=\dfrac{y}{3}$$; this latter form is called the inverse function to the one originally given.

If $$y=3x$$, $$\dfrac{dy}{dx}=3$$; if $$x=\dfrac{y}{3}$$, $$\dfrac{dx}{dy} = \dfrac{1}{3}$$, and we see that

Consider $$y=4x^2$$, $$\dfrac{dy}{dx}=8x$$; the inverse function is