Introduction to Calculus
May 30, 2017
- Exercises, May 30, 2017
- I
- Differentiate the function $f(x)$ provided the basic rules
\begin{align}
\left(x^n\right)'&=nx^{n-1}&(n=1,2,3,\dots)\\
\left(\frac 1{x^n}\right)'
&=-\frac n{x^{n+1}}&(n=1,2,3,\dots)\\
\left(\sqrt{x}\right)'&=
\frac 12\cdot \frac 1{\sqrt x}&
\end{align}
-
(1)
$f(x)=\frac 1{x+2}$
(2)
$f(x)=\frac {x+3}{x-1}$
(3)
$f(x)=\frac 1 {2x+1}$
(4)
$f(x)=\frac x{2x-1}$
(5)
$f(x)=\frac 1{x^2+1}$
(6)
$f(x)=\frac {x+1}{x^2+1}$
(7)
$f(x)=\frac {x^2}{x-1}$
(8)
$f(x)=x^2\sqrt{x}$
(9)
$f(x)=\frac 1{x\sqrt{x}}$
(10)
$f(x)=\frac 1{x^2\sqrt{x}}$
(11)
$f(x)=\frac {x^2}{x^2+1}$
(12)
$f(x)=\frac x{x^2+x+1}$
- II
- Differentiate the function $f(x)$.
-
(1)
$f(x)=\frac 1{(3x+1)^3}$
(2)
$f(x)=(1-2x)^5$
(3)
$f(x)=\left(\frac {x-1}x\right)^5$
(4)
$f(x)=\left(3-2x^2\right)^3$
(5)
$f(x)=\sqrt{x-1}$
(6)
$f(x)=\frac 1{\sqrt{x-1}}$
(7)
$f(x)=\frac 1{\sqrt{x^2+x+1}}$
(8)
$f(x)=\frac x{\sqrt{1-x^2}}$
(9)
$f(x)=\frac x{\sqrt{1+x^2}}$