Page:Calculus Made Easy.pdf/277

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

(2) $$\dfrac{x}{\sqrt{ x^2 + a^2}}$$.

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

(4) $$\dfrac{ax}{\sqrt{(a - x^2)^3}}$$.

(5) $$\dfrac{2a^2 - x^2}{x^3 \sqrt{ x^2 - a^2}}$$.

(6) $$\dfrac{\frac{3}{2} x^2 \left[ \frac{8}{9} x \left( x^3 + a \right) - \left( x^4 + a \right) \right]}{(x^4 + a)^{\frac{2}{3}} (x^3 + a)^{\frac{3}{2}}}$$.

(7) $$\dfrac{2a \left(x - a \right)}{(x + a)^3}$$.

(8) $$\frac{5}{2} y^3$$.

(9) $$\dfrac{1}{(1 - \theta) \sqrt{1 - \theta^2}}$$.

(1) $$\dfrac{dw}{dx} = \dfrac{3x^2 \left( 3 + 3x^3 \right)} {27 \left(\frac{1}{2} x^3 + \frac{1}{4} x^6 \right)^3}$$.

(2) $$\dfrac{dv}{dx} = - \dfrac{12x}{\sqrt{1 + \sqrt{2} + 3x^2} \left(\sqrt{3} + 4 \sqrt{1 + \sqrt{2} + 3x^2}\right)^2}$$.

(3) $$\dfrac{du}{dx} = - \dfrac{x^2 \left(\sqrt{3} + x^3 \right)} {\sqrt{ \left[ 1 + \left( 1 + \dfrac{x^3}{\sqrt{3}} \right) ^2 \right]^3}}$$.

(2) $$1.44$$.

(4) $$\dfrac{dy}{dx}=3x^2+3$$; and the numerical values are: $$3$$, $$3\frac{3}{4}$$, $$6$$, and $$15$$.

(5) $$\pm \sqrt{2}$$.

(6) $$\dfrac{dy}{dx} = - \dfrac{4}{9} \dfrac{x}{y}$$. Slope is zero where $$x=0$$; and is $$\mp \dfrac{1}{3 \sqrt{2}}$$ where $$x=1$$.

(7) $$m=4$$, $$n=-3$$.

(8) Intersections at $$x=1$$, $$x=-3$$. Angles $$153^\circ \;26'$$, $$2^\circ \;28'$$.

(9) Intersections at $$x=3.57$$, $$x=3.50$$. Angles $$16^\circ \;16'$$.

(10) $$x=\tfrac{1}{3}$$, $$y=2\tfrac{1}{3}$$, $$b=-\tfrac{5}{3}$$.