1
Spot and correct the error(s) in the following calculation.
\begin{align*}
f(x)&=\frac{2x}{x+1}\\
f'(x)&=\frac{2(x+1)+2x}{(x+1)^2}\\
&=\frac{2(x+1)}{(x+1)^2}\\
&=\frac{2}{x+1}
\end{align*}
Subsection 2.6.2 Exercises
Exercises — Stage 1
2
True or false: \(\ds\diff{}{x}\{2^x\}=x2^{x-1}\text{.}\)
Exercises — Stage 2
3
Differentiate \(f(x)=\frac{2}{3}x^6+5x^4+12x^2+9\) and factor the result.
4
Differentiate \(s(t)=3t^4+5t^3-\frac{1}{t}\text{.}\)
5
Differentiate \(x(y) = \left(2y+\frac{1}{y}\right)\cdot y^3\text{.}\)
6
Differentiate \(T(x) = \dfrac{\sqrt{x}+1}{x^2+3}\text{.}\)
7 (✳)
Compute the derivative of \(\left(\dfrac{7x+2}{x^2+3}\right)\text{.}\)
8
What is \(f'(0)\text{,}\) when \(f(x)=(3x^3+4x^2+x+1)(2x+5)\text{?}\)
9
Differentiate \(f(x)=\dfrac{3x^3+1}{x^2+5x}\text{.}\)
10 (✳)
Compute the derivative of \(\left(\dfrac{3x^2+5}{2-x}\right)\)
11 (✳)
Compute the derivative of \(\left(\dfrac{2-x^2}{3x^2+5}\right)\text{.}\)
12 (✳)
Compute the derivative of \(\left(\dfrac{2x^3+1}{x+2}\right)\text{.}\)
13 (✳)
For what values of \(x\) does the derivative of \(\dfrac{\sqrt{x}}{1-x^2}\) exist? Explain your answer.
14
Differentiate \(f(x)=\left(3\sqrt[5]{x}+15\sqrt[3]{x}+8\right)\left(3x^2+8x-5\right)\text{.}\)
15
Differentiate \(f(x)=\dfrac{(x^2+5x+1)(\sqrt{x}+\sqrt[3]{x})}{x}\text{.}\)
16
Find all \(x\)-values where \(f(x)=\dfrac{1}{\frac{1}{5}x^5+x^4-\frac{5}{3}x^3}\) has a horizontal tangent line.
Exercises — Stage 3
17 (✳)
Find an equation of a line that is tangent to both of the curves \(y = x^2\) and \(y = x^2 - 2x + 2\) (at different points).
18
[1998H] Find all lines that are tangent to both of the curves \(y=x^2\) and \(y=-x^2+2x-5\text{.}\) Illustrate your answer with a sketch.
19 (✳)
Evaluate \(\displaystyle \lim_{x\to 2}\left( \dfrac{x^{2015}-2^{2015}}{x-2}\right).\)