Derivatives

Rate of Change

Consider the function \( f(x)=3x+7 \). We have that \( f(5)=3\cdot 5+7=22 \). The input value is \( 5 \), and the output is \( 22 \). If we increase the input by \( 2 \), it becomes \( 7 \) and the output becomes \( 28 \). This means if the input increases by \( 2 \), the output increases by \( 6 \).

Assume now that the original input is increased by \( 12 \). It becomes \( 17 \). The output is \( f(17)=3\cdot 17+7=58 \), which is by \( 36 \) bigger than the original output.

In general, if the input is increased by value \( t \), the output increases by \( 3\cdot t \), and this number \( 3 \) is called the rate of change.

We can study only the rate of change of a function. Let us consider the following example

Example 1
 
Consider the function \( f(x)=5x+1 \). Find the rate of change of \( f \).

Now we may think that we are ready to define the rate of change for a general function. However, the following example is quite depressing. Consider the function \( g(x)=x^2 \).

Let’s start with \( x_O=3 \). We first find \( g(x_O)=9 \). If we take \( x_N=4 \), then \( g(x_N)=16 \) and the change in \( g \) is \( \frac{16-9}{4-3}=7 \) times bigger than the change in \( x \).

However, if we take \( x_{N^{\prime}}=5 \) we get \( g(x_{N^{\prime}})=25 \) and the change in \( g \) is \( \frac{25-9}{5-3}=8 \) times bigger than the change in \( x \).

If we take \( x_{N^{\prime\prime}}=500 \), then \( g(X_{N^{\prime\prime}})=250000 \) and the change in \( g \) is \( \frac{250000-9}{500-3}=\frac{249999}{497} \) times bigger than the change in \( x \).

Despite this fact we are inclined to say that the rate of change of the function \( f(x)=x^2 \) is small. This is because nobody cares about the difference between \( g(500) \) and \( g(3) \).

We are interested in the infinitesimal rate of change, and this is called a derivative of the function.

The definition

Definition of derivative
 

Assume that \( f \) is a function, and \( a \) a real number that belongs to the domain of definition of \( f \). If the limit \( \lim_{h\to 0}\frac{f(a+h)-f(a)}h \) exists, we say that the function \( f \) is differentiable at point \( a \) and we called the previous limit the derivative of \( f \) at point \( a \). We denote the derivative by \( f^{\prime}(a) \), i.e.:

\[ f^{\prime}(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}h.\]

The slope of the tangent

slope of a tangent

If we draw the graph of the function \( f \) in the \( xy \) plane, then the slope of the line between the points \( (x,f(x)) \) and \( (x+\Delta x, f(x+\Delta x)) \) is equal to \( \frac{f(x+\Delta x)-f(x)}{\Delta x} \).

As \( \Delta x \) gets smaller and smaller, the line between \( (x,f(x)) \) and \( (x+\Delta x, f(x+\Delta x)) \) becomes a more accurate approximation to the tangent line of the graph of \( f \). The quantity \[ \lim_{\Delta x\to0}\frac{f(x+\Delta x)-f(x)}{\Delta x)}\] corresponds to the slope of the tangent line to the graph of \( f \) at the point \( (x,f(x)) \).

Derivatives of \( x^m \), \( e^x \), \( \sin x \), and \( \cos x \)

Our first theorem states that derivative of a constant function is \( 0 \).

Theorem (Derivative of a constant)
 
Let \( C \) be arbitrary real number, and let \( f \) be the function defined as \( f(x)=C \) for all \( x\in\mathbb R \). Then \( f \) is differentiable for any \( a\in\mathbb R \) and \( f^{\prime}(a)=0 \).

Our next theorem states that the derivative of a function of the form \( f(x)=x^m \) is \( f^{\prime}(x)=mx^{m-1} \) if \( m \) is a positive integer.

Theorem (Power rule)
 
Let \( m\in\mathbb N \) and let \( f(x)=x^m \). Then \( f^{\prime}(a)=ma^{m-1} \) for all \( a\in\mathbb R \).

The power rule holds even for \( m\in\mathbb R \), but the proof is more complicated, and we will omit it for now.

Theorem (General power rule)
 
Let \( m\in\mathbb R \) and let \( f(x)=x^m \). Then \( f^{\prime}(a)=ma^{m-1} \) for all \( a\in\mathbb R \).

Theorem (Derivative of the exponential function)
 
If \( f(x)=e^x \), then \( f^{\prime}(a)=e^a \) for every real number \( a \).

Theorem (Derivatives of \( \sin \) and \( \cos \))
 
If \( f(x)=\sin x \), then \( f^{\prime}(a)=-\cos a \) for every real number \( a \). If \( g(x)=\cos x \), then \( g^{\prime}(a)=-\sin a \).


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