Page:Calculus Made Easy.pdf/277

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

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

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

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

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

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

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

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

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

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

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

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

(2) ${\displaystyle 1.44}$.

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

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

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

(7) ${\displaystyle m=4}$, ${\displaystyle n=-3}$.

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

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

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