In this exercise we test in particular the following rules for differentiation: The product rule and the rule for a composite function (also called the chain rule). And of course the derivative of $\,\cos\,,$$\,\sin\,,$$\,\ln\,$ og $\,\mathrm{exp}\,.$
Note that the fraction $\,\frac{\e^x}{x^2}\,$ can be treated as $\,\e^x \cdot\frac 1{x^2}\,,$ i.e. and can be differentiated following the product rule.
If the function $\,y=f(x)\,$ has an inverse function $\,x=f^{-1}(y)\,$ we can find the derivative of $\,f^{-1}(y)\,$ like this:
$$(f^{-1})'(y)=\frac{1}{f'(x)}\,.$$
C
Sine has in the interval $\,\left[-\frac{\pi}2\,,\,\frac{\pi}2\,\right]\,$ an inverse function denoted $\arcsin\,.$
Determine the derivative of $\arcsin\,.$
hint
$\,\arcsin’(y)=\ldots\,$
hint
In addition you will probably need the idiot rule.
Show that the differential quotient of tangent in every $\,t\,$ in the domain is given by the expression
$$\tan'(t)=\,1+\tan^2(t)\,.$$
E
Tangent has in the open interval $\,]-\frac{\pi}2\,,\,\frac{\pi}2\,[\,$ and inverse function arcustangent that is denoted $\arctan\,.$ Determine $\,\mathrm{arctan}’(x)\,$ for every $\,x\,$ in the interval.
answer
$$\frac{1}{1+x^2}$$
Exercise 6: Differential Quotient from the Definition
F
Show that the real function $\,f\,$ given by $\,f(x)=x^2\,$ is differentiable in every $\,x0\in \reel$ with the differential quotient
$$\frac{d}{dx} f(x_0)=2x_0\,.$$
Hint: Substitute the expressionf for $\,f\,$ in the expression
$$f(x)=f(x_0)+a(x-x_0)+\epsilon(x-x_0)(x-x_0)\,$$
and isolate $\,\epsilon(x-x_0)\,.$
Exercise 7: Quadratic Equations with Real Coefficients. Exercise
G
Solve The equations below both within $\reel$ and within $\mathbb C\,.$
$\,2x^2+9x-5=0$
$\,x^2-4x=0$
$\,x^2-4x+13=0$
answer
First equation: Within both set of numbers the solutions are $\,\displaystyle{-5\,\,\,\mathrm{and}\,\,\,\frac 12}\,.$
Second equation: Within both set of numbers the solutions are $\,\displaystyle{0\,\,\,\mathrm{and}\,\,\,4}\,.$
Third equation: Has no real solutions. Within the complex numbers it has the solutions $\,\displaystyle{2-3i\,\,\,\mathrm{and}\,\,\,2+3i}\,.$
H
Solve the equation
$$
2(x+1-i)(x+1+i)=0$$
and show that it is of the type quadratic equation with real coefficients.
answer
$x=-1+i$ and $x=-1-i$
Exercise 8: Quadratic Equations with Complex Coefficients (Advanced)
Show that the function:
\begin{equation}
f_{1}(x)=
\begin{cases}
|x|/x& \text{if}\, x \neq 0 \
0& \text{if}\, x = 0
\end{cases}
\end{equation}
is not an epsilon function.
L
Show that the function:
\begin{equation}
f_{2}(x)=1-\cos(x)
\end{equation}
is an epsilon function.
M
Show that the complex function:
\begin{equation}
f_{3}(x)=i\e^{ix}-i
\end{equation}
is an epsilon function.
%# Exercise 10: Inverse Functions and Their Differential Quotients
%####### begin:question
%Explain that the function $\,f\,$ given by $\,f(x)=x^3+x^2+x+1\,$ has an inverse function $\,f^{-1}\,$ defined for all of $\reel\,.$ Determine the differential quotient of $f^{-1}$ in the numbers $\,x_0=4\,$ and $\,x_1=0\,$.
%####### end:question
%####### begin:answer
%$ 1.\quad$ Because $\,f\,$ is monotonous and therefore injective on $\,\reel\,.$
%Two new functions $\,\cosh\,$ and $\,\sinh\,$ are definered by the expressions below. The first is called hyperbolic cosine and the other hyperbolsk sine.
%
%####### begin:question
%Show directly from the definition of the two functions that:
%
$$
%\cosh^2(x)-\sinh^2(x)=1\,.
%$$
%####### end:question
%####### begin:question
%The words cosine and sine are parts of the name of the functions. Name some of the characteristics about the hyperbolic functions and cosine and sine.
%####### end:question
Differentiate $\,(x^2+7)^{13}\,$.