1.1 Einführung zur Differentialrechnung
Aus Online Mathematik Brückenkurs 2
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Content:
- Derivative definition (overview).
- Derivative of \displaystyle x^\alpha, \displaystyle \ln x, \displaystyle e^x, \displaystyle \cos x, \displaystyle \sin x and \displaystyle \tan x.
- Derivative of sums and differences.
- Tangents and normals to curves.
Learning outcomes:
After this section, you will have learned: :
- That the first derivative \displaystyle f^{\,\prime}(a) is the slope of the curve \displaystyle y=f(x) at the point \displaystyle x=a.
- That the first derivative is the instantaneous rate of change of a quantity (such as speed, price increase, and so on.).
- That there are functions that are not differentiable (such as \displaystyle f(x)=\vert x\vert at \displaystyle x=0).
- To differentiate \displaystyle x^\alpha, \displaystyle \ln x, \displaystyle e^x, \displaystyle \cos x, \displaystyle \sin x and \displaystyle \tan x as well as the sums / differences of such terms.
- To determine the tangent and normal to the curve \displaystyle y=f(x).
- That the derivative can be denoted by \displaystyle f^{\,\prime}(x) och \displaystyle df/dx(x).
Introduction
When studying mathematical functions and their graphs one of the main areas of study of a function is the way it changes, i.e. whether a function is increasing or decreasing and the rate at which this is taking place.
One makes use of the concept rate of change (or speed of change), which is a measure of how the value of the function (\displaystyle y) is changing per unit increase in the variable (\displaystyle x). If one knows two points on a graph of a function one can get a measure of the rate of change of the function between these points by calculating the increment ratio
- REDIRECT Template:Abgesetzte Formel
Example 1
The linear functions \displaystyle f(x)=x and \displaystyle g(x)=-2x change by the same amount everywhere. Their rates of change are \displaystyle 1 and. \displaystyle −2, which are the slopes of these straight lines.
| 1.1 - Figur - Grafen till f(x) = x | 1.1 - Figur - Grafen till f(x) = -2x | |
| Graph of f(x) = x has slope 1. | Graph of g(x) = - 2x has slope - 2. |
Thus for a linear function the rate of change is the same as the slope.
For a function for which the value changes with time, it is natural to use the term speed of change because rate of change specifies how the value of the functions is changing per unit time.
If a car is moving at a speed of 80 km/h then the distance traveled, s km, after t hours is given by the function \displaystyle s(t)=80 t. The rate of change of the function indicates how the value of the function is changing per hour, which of course is the same as the car's speed, 80 km/h.
For non-linear functions however, the slope of the curve of the function is changing all the time and thus also the function's rate of change. In order to determine how such a function is changing, we can either give the functions average change ( its mean) between two points on the curve of the function, or the instantaneous rate of change at one point on the curve.
Example 2
For the function \displaystyle f(x)=4x-x^2 one has \displaystyle f(1)=3, \displaystyle f(2)=4 and \displaystyle f(4)=0.
- Mean change (mean slope) from \displaystyle x = 1
to \displaystyle x = 2 is
- REDIRECT Template:Abgesetzte Formel
- Mean change from \displaystyle x = 2 to \displaystyle x = 4 is
- REDIRECT Template:Abgesetzte Formel
- Between \displaystyle x = 1 and \displaystyle x = 4 the mean is
- REDIRECT Template:Abgesetzte Formel
| 1.1 - Figur - Medelförändring för f(x) = x(4 - x) mellan x = 1 och x = 2 | 1.1 - Figur - Medelförändring för f(x) = x(4 - x) mellan x = 1 och x = 4 | |
| Between x = 1 and x = 2 the function has the mean 1/1 = 1. | Between x = 1 and x = 4 the function has the mean (-3)/3 = -1. |
Definition of the derivative
To calculate the instantaneous rate of change of a function, that is. the slope of its curve at a point P, we temporarily use an additional point Q in the vicinity of P and construct the increment ratio between P and Q:
Increment ratio
- REDIRECT Template:Abgesetzte Formel
If we allow Q to approach P (that is allow \displaystyle h \rightarrow 0) we can work out what the value would be if the points coincided, and thus obtain the slope at the point P. We call this value for the derivative of \displaystyle f(x) at the point P, and can be interpreted as the instantaneous rate of change of \displaystyle f(x) at the point P.
The derivative of a function \displaystyle f(x) is written as \displaystyle f^{\,\prime}(x) and may be formally defined as follows:
The derivative of a function \displaystyle f(x), is defined as
- REDIRECT Template:Abgesetzte Formel
If \displaystyle f^{\,\prime}(x_0) exists, one says that \displaystyle f(x) is differentiable at the point \displaystyle x=x_0.
Different notations for the derivative are used, for example,
| Function | derivative |
|---|---|
| \displaystyle f(x) | \displaystyle f^{\,\prime}(x) |
| \displaystyle y | \displaystyle y^{\,\prime} |
| \displaystyle y | \displaystyle Dy |
| \displaystyle y | \displaystyle \dfrac{dy}{dx} |
| \displaystyle s(t) | \displaystyle \dot s(t) |
The sign of the derivative
The derivatives sign (+/−) tells us if the functions graph slopes upwards or downwards, that is, if the function is increasing or decreasing:
- \displaystyle f^{\,\prime}(x) > 0 ( (positive slope) means that \displaystyle f(x) is increasing.
- \displaystyle f^{\,\prime}(x) < 0 (negative slope) means that \displaystyle f(x) is decreasing.
- \displaystyle f^{\,\prime}(x) = 0 (no slope) means thatt \displaystyle f(x) is stationary (horisontal).
Example 3
- \displaystyle f(2)=3\ means that 'the function value' is \displaystyle 3 at \displaystyle x=2.
- \displaystyle f^{\,\prime}(2)=3\ means that 'the derivatives value' is \displaystyle 3 when \displaystyle x=2, which in turn means that the functions graph has a slope \displaystyle 3 at \displaystyle x=2.
Example 4
From the figure one can obtain that
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\displaystyle \begin{align*} f^{\,\prime}(a) &> 0\\[4pt] f(b) &= 0\\[4pt] f^{\,\prime}(c) &= 0\\[4pt] f(d) &= 0\\[4pt] f^{\,\prime}(e) &= 0\\[4pt] f(e) &< 0\\[4pt] f^{\,\prime}(g) &> 0 \end{align*} |
1.1 - Figur - Grafen y = f(x) med punkter x = a, b, c, d, e och g |
Note the meaning of \displaystyle f(x) respektive \displaystyle f^{\,\prime}(x).
Example 5
The temperature in a thermos is given by a function, where \displaystyle T(t) is the temperature of the thermos after \displaystyle t minutes. Interpret the following using mathematical symbols:
- After 10 minutes the temperature is 80°.
\displaystyle T(10)=80 - After 2 minutes the temperature is dropping in the thermos by 3° per minute
\displaystyle T'(2)=-3 (the temperature is decreasing, which is why the derivative is negative)
Example 6
The function \displaystyle f(x)=|x| does not have a derivative at \displaystyle x=0. One cannot determine how the graph of the function slopes at the point \displaystyle (0,0) (see figure below).
One can express this, for example, in one of the following ways:"\displaystyle f^{\,\prime}(0) does not exist", "\displaystyle f^{\,\prime}(0) is not defined " or "\displaystyle f(x) is not differentiable at \displaystyle x=0".
Differentiation rules
Using the definition of differentiation one can determine the derivatives for the standard types of functions.
Example 7
If \displaystyle f(x)=x^2 then, according to the definition of the increment ratio
- REDIRECT Template:Abgesetzte Formel
If we then let \displaystyle h go to zero, we see that the slope at the point becomes \displaystyle 2x. We have thus shown that the slope of an arbitrary point on the curve \displaystyle y=x^2 is \displaystyle 2x, That is the derivative of \displaystyle x^2 is \displaystyle 2x.
In a similar way, we can deduce general differentiation rules :
| Function | Derivative |
|---|---|
| \displaystyle x^n | \displaystyle nx^{n-1} |
| \displaystyle \ln x | \displaystyle 1/x |
| \displaystyle e^x | \displaystyle e^x |
| \displaystyle \sin x | \displaystyle \cos x |
| \displaystyle \cos x | \displaystyle -\sin x |
| \displaystyle \tan x | \displaystyle 1/\cos^2 x |
In addition, for sums and differences of expressions of functions one has
- REDIRECT Template:Abgesetzte Formel
Additionally, if k is a constant, then
- REDIRECT Template:Abgesetzte Formel
Example 8
- \displaystyle D(2x^3 - 4x + 10 - \sin x)
= 2\,D x^3 - 4\,D x + D 10 - D \sin x
\displaystyle \phantom{D(2x^3 - 4x + 10 - \sin x)}{} = 2\cdot 3x^2 - 4\cdot 1 + 0 - \cos x - \displaystyle y= 3 \ln x + 2e^x \quad gives that \displaystyle \quad y'= 3 \cdot\frac{1}{x} + 2 e^x = \frac{3}{x} + 2 e^x\,.
- \displaystyle \frac{d}{dx}\Bigl(\frac{3x^2}{5} - \frac{x^3}{2}\Bigr) = \frac{d}{dx}\bigl(\tfrac{3}{5}x^2 - \tfrac{1}{2}x^3\bigr) = \tfrac{3}{5}\cdot 2x - \tfrac{1}{2}\cdot 3x^2 = \tfrac{6}{5}x - \tfrac{3}{2}x^2\,.
- \displaystyle s(t)= v_0t + \frac{at^2}{2} \quad gives that \displaystyle \quad s'(t)=v_0 + \frac{2at}{2} = v_0 + at\,.
Example 9
- \displaystyle f(x) = \frac{1}{x} = x^{-1} \quad gives that \displaystyle \quad f^{\,\prime}(x) = -1 \cdot x^{-2} = -\frac{1}{x^2}\,.
- \displaystyle f(x)= \frac{1}{3x^2} = \tfrac{1}{3}\,x^{-2} \quad gives that \displaystyle \quad f^{\,\prime}(x) = \tfrac{1}{3}\cdot(-2)x^{-3} = -\tfrac{2}{3} \cdot x^{-3} = -\frac{2}{3x^3}\,.
- \displaystyle g(t) = \frac{t^2 - 2t + 1}{t} = t -2 + \frac{1}{t} \quad gives that\displaystyle \quad g'(t) = 1 - \frac{1}{t^2}\,.
- \displaystyle y = \Bigl( x^2 + \frac{1}{x} \Bigr)^2
= (x^2)^2 + 2 \cdot x^2 \cdot \frac{1}{x} + \Bigl(\frac{1}{x} \Bigr)^2
= x^4 + 2x + x^{-2}
\displaystyle \qquad\quad gives that\displaystyle \quad y' =4x^3 + 2 -2x^{-3} = 4x^3 + 2 - \frac{2}{x^3}\,.
Example 10
The function \displaystyle f(x)=x^2 + x^{-2} has the derivative
- REDIRECT Template:Abgesetzte Formel
This means, for example, that \displaystyle f^{\,\prime}(2) = 2\cdot 2 - 2/2^3= 4- \frac{1}{4} = \frac{15}{4} and that \displaystyle f^{\,\prime}(-1) = 2 \cdot (-1) - 2/(-1)^3 = -2 + 2 = 0. However, the derivative \displaystyle f'(0) is not defined.
Example 11
An object moves according to \displaystyle s(t) = t^3 -4t^2 +5t, where \displaystyle s(t) km is the distance from the starting point after \displaystyle t hours. Calculate \displaystyle s'(3) and explain what the value stands for.
Differentiating with respect to the time
- REDIRECT Template:Abgesetzte Formel
This might suggest that after 3 hours the object's speed is 8 km / h.
Example 12
The total cost \displaystyle T dollars for the manufacture of \displaystyle x objects is given by the function
- REDIRECT Template:Abgesetzte Formel
Calculate and explain the meaning of the following expressions.
- \displaystyle T(120)
\displaystyle T(120)=40000 + 370 \cdot 120 - 0{,}09 \cdot 120^2 = 83104\,.
The total cost to manufacture 120 objects is 83104 dollars. - \displaystyle T'(120)
The derivative is given by \displaystyle T^{\,\prime}(x)= 370 - 0{,}18x and therefore, is- REDIRECT Template:Abgesetzte Formel
Tangents and normals
A tangent to a curve is a straight line tangential to the curve.
A normal to a curve at a point on the curve is a straight line that is perpendicular to the curve at the point (and hence perpendicular to the curve's tangent at this point).
For perpendicular lines, the product of their slopes is \displaystyle –1, i.e. if the tangents slope is \displaystyle k_T and the normals is \displaystyle k_N then \displaystyle k_T \cdot k_N = -1. Since we can determine the slope of a curve with the help of the derivative, we can also determine the equation of a tangent or a normal, if we know the equation for the curve.
Example 13
Determine the equation for the tangent and the normal to the curve \displaystyle y=x^2 + 1 at the point \displaystyle (1,2).
We write the tangents equation as \displaystyle y = kx + m. Since it is to tangent (touch) the curve at \displaystyle x=1 it must have a slope of \displaystyle k= y'(1), i.e.
- REDIRECT Template:Abgesetzte Formel
The tangent also passes through the point \displaystyle (1,2) and therefore \displaystyle (1,2) must satisfy the tangents equation
- REDIRECT Template:Abgesetzte Formel
The tangents equation is thus \displaystyle y=2x.
The slope of the normal is \displaystyle k_N = -\frac{1}{k_T} = -\frac{1}{2} .
In addition, the normal also passes through the point \displaystyle (1, 2) , i.e.
- REDIRECT Template:Abgesetzte Formel
The normal has the equation \displaystyle y= -\frac{x}{2} + \frac{5}{2} = \frac{5-x}{2}.
| 1.1 - Figur - Tangentlinjen y = 2x | 1.1 - Figur - Normallinjen y = (5 - x)/2 | |
| Tangent \displaystyle y=2x | Normal \displaystyle y=(5-x)/2 |
Example 14
The curve \displaystyle y = 2 \, e^x - 3x has a tangent with a slope of \displaystyle –1. Determine the point of tangency (where the tangent touches the curve).
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The derivative of the right-hand side is \displaystyle y' = 2 \, e^x -3 and at the point of tangency the derivative must be equal to \displaystyle -1, that is, \displaystyle y' = -1, and this gives us the equation
which has a solution \displaystyle x=0. At the point \displaystyle x=0 the curve has \displaystyle y-value \displaystyle y(0) = 2 \, e^0 - 3 \cdot 0 = 2 and therefore the point of tangency is \displaystyle (0,2). | 1.1 - Figur - Kurvan y = 2e^x - 3x och tangentlinjen genom (0,2) |
