Differentiation.

It's a warm summer evening in ancient Greece...

::: author Apollonius of Perga A tangent to a curve C is a line l that intersects C, such that no other line could fall between l and C. :::

Finding the tangent Using a limit.

• the line through P an Q is called a secant.
• the tangent to the curve (at P) is the limit of the secant line for Q approaching P.

• the slope of the secant through P and Q is:

$$\frac{f(x_0+h)-f(x_0)}{h}$$

• The slope of the tangent to C at P is

$$\underset{h\to 0}{lim}\frac{f(x_0+h)-f(x_0)}{h}$$

Differentiability and continuity

Definition

Let f be a function defined on an open interval I, and let $x_0\in I$ the derivative of f at $x_0$ is

$$f^{\prime }(x_0)=li{m}_{h\to 0}\frac{f(x_0+h)-f(x_0)}{h}$$

provided this limit exists.

• If the limit exists we say: f is differentiable at $x_0$

• When a function is differentiable it is also continuous.

• But a function can be continuous but not differentiable.

Possible notations for derivatives:

• $f^{\prime }(x_0)$
• $\frac{df}{dx}(x_0)$
• $f_x(x_0)$

Differentiation Rules 📜

• Sum Rule: $\frac{d}{dx}(f+g)(x)=f^{\prime }(x)+g^{\prime }(x)$
• Difference Rule: $\frac{d}{dx}(f-g)(x)=f^{\prime }(x)-g^{\prime }(x)$
• Constant multiple Rule: $\frac{k\cdot f}{x}=k\cdot f^{\prime }(x)$
• Product Rule: $\frac{d}{dx}(f\cdot g)(x)=f^{\prime }(x)\cdot g(x)+f(x)g^{\prime }(x)$
• Quotient Rule: $\frac{d}{dx}(\frac{f}{g})(x)=\frac{f^{\prime }(x)\cdot g(x)-f(x)\cdot g^{\prime }(x)}{g(x{)}^2}$
• Chain Rule: $\frac{d}{dx}f(g(x))=f^{\prime }(g(x))\cdot g^{\prime }(x)$

Derivatives of elementary functions:

$f(x)$	$f^{\prime }(x)$	condition
c	0	$c\in \mathbb{R}$
$\sqrt{x}$	$\frac{1}{2\sqrt{x}}$	$x>0$
$\frac{1}{x}$	$-\frac{1}{x^2}$	$x\ne 0$
$x^{\alpha }$	$\alpha x^{\alpha -1}$	$\alpha \in \mathbb{R}$
$e^x$	$e^x$
$\ln x$	$\frac{1}{x}$	$x>0$
$\sin x$	$\cos x$
$\cos x$	$-\sin x$
$\arcsin x$	$\frac{1}{\sqrt{1-x^2}}$	$\text{abs}x<1$
$\arccos x$	$-\frac{1}{\sqrt{1-x^2}}$	$\text{abs}x<1$
$\arctan x$	$\frac{1}{\sqrt{x^2+1}}$

The tangent Line:

info

Let f be differentiable at a. The tangent line to the graph of f at P(a, f(a)) is given by the equation:

$$y=f(a)+f^{\prime }(a)(x-a)$$

Linearization

the function L is called the linearization of f at a. (it's the tangent line)

$$L(x)=f(a)+f^{\prime }(a)(x-a)$$

Linearization as approximation:

• linearizations can be used as an approximation for a more complicated function.
• linearizations have less error the closer x is to the point at which you linearized (a in the tangent equation)

Extreme Values:

Definition

Let f be a real function with domain D. Let $c\in D$.

• f has an absolute maximum value on D at c if:

$$\mathrm{\forall }x\in D[f(x)\le f(c)]$$

• f has an absolute minimum value on D at c if:

$$\mathrm{\forall }x\in D[f(x)\ge f(c)]$$

Extreme Value info

Conditions:

• function f is continuous on interval [a, b]
• interval [a, b] is closed == (a and b are included in the interval)
• interval [a, b] is bounded == (a and b are real numbers)

Result:

• f attains both an absolute maximum value and an absolute minimum value on [a, b].

Local Extrema:

Definition

Let f be a real function with domain D. Let $c\in D$.

• f has an local maximum value at c if there exists an open interval I containing c such that:

$$\mathrm{\forall }x\in I\cap D[f(x)\le f(c)]$$

• f has an local minimum value at c if there exists an open interval $I\subseteq D$:

$$\mathrm{\forall }x\in I\cap D[f(x)\ge f(c)]$$

First Derivative info

If f has a local maximum or minimum value at an interior point c of it's domain, and f' is defined at c, then f'(c) = 0

If: $f:I\to R$ has a local extreme value at c: The requirements of the info do not hold:

• c is not an interior point of I
• f is not differentiable at c. Or:
• $f^{\prime }(c)=0$.

Critical points

Definition

An interior point of the domain of f where f' is undefined or where f'(c) = 0 is called a critical point of f.

• A critical point is not necessarily a local extreme.

Finding absolute extrema:

1. Find all points (a, b) where f' does not exist.
2. Find all $c\in (a,b)$ where f'(c) = 0.
3. Calculate the f(x) for x = [a, b, points that don't exist, points c where f'(c) = 0]
4. The largest value is the absolute maximum value.
5. The smallest value is the absolute minimum value.

L'Hôpital's rule

Indeterminate form $\frac{0}{0}$

L'Hôpital's rule

Suppose that f(c) = g(c) = 0, that f and g are differentiable on an open interval I containing c and that $g^{\prime }(x)\ne 0$ on I if $x\ne c$,

$$\underset{x\to c}{lim}\frac{f(x)}{g(x)}=\underset{x\to c}{lim}\frac{f^{\prime }(x)}{g^{\prime }(x)}$$

assuming that the limit on the right side of the equation exists.

• if the limit on the right hand side is of type $\frac{0}{0}$ you can apply L'Hôpital again.

Indeterminate form $\frac{\mathrm{\infty }}{\mathrm{\infty }}$

• L'Hôpital's rule is also correct when $c=\mathrm{\infty }\text{ or }c=-\mathrm{\infty }$

Indeterminate form $0\cdot \mathrm{\infty }$

For limits of the type: $0\cdot \mathrm{\infty }$ you can write the limit as a fraction to be able to apply L'Hôpital.

• $\underset{x\to c}{lim}f(x)\cdot g(x)=\underset{x\to c}{lim}\frac{f(x)}{\frac{1}{g(x)}}$
• $\underset{x\to c}{lim}f(x)\cdot g(x)=\underset{x\to c}{lim}\frac{g(x)}{\frac{1}{f(x)}}$

Indeterminate form $0^{\mathrm{\infty }}$ and similar

If a limit of type $\underset{x\to c}{lim}f(x{)}^{g(x)}$ after substitution results in a limit of type $0^{\mathrm{\infty }},{\mathrm{\infty }}^0\text{ or }{1}^{\mathrm{\infty }}$ then it should be written as:

$$\underset{x\to c}{lim}f(x{)}^{g(x)}=\underset{x\to c}{lim}{e}^{g(x)\ln f(x)}$$

• first calculate $L=\underset{x\to c}{lim}g(x)\ln f(x)$ (hopital may be needed at this step)
• then $\underset{x\to c}{lim}f(x{)}^{g(x)}=e^L$