Examples for derivatives – Serlo

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In this chapter we want to summarise the most important examples of derivatives. The derivative rules will allow us for computing derivatives of composite functions.

Table of important derivatives[Bearbeiten]

In the following table , and is given. We also define , and .

function term term of the derivative function domain of definition of the derivative

Examples for computing derivatives[Bearbeiten]

Now we will calculate some examples of derivatives from the table above. Often it comes down to determining the differential quotient of the function, i.e. a limit value. But sometimes it is also useful to use the calculation rules from the chapter before.

Constant functions[Bearbeiten]

We start with some simple derivatives:

Theorem (Derivative of a constant function)

Every constant function is differentiable on all of with derivative .

Proof (Derivative of a constant function)

Let . Then there is

Power functions with natural numbers as powers[Bearbeiten]

Now we turn to the derivative of power functions with natural powers. First we will deal with a few special cases:

Example (Derivative of the identity and the square function)

The functions

and

are differentiable on all of . Further there is for :

as well as

For the derivative of we used the 3rd binomial formula .

Exercise (Derivative of a power function)

Compute die derivative von

Solution (Derivative of a power function)

For there is

Instead of using the identity , we could also have calculated using polynomial division.

Now we turn to the general case, i.e. the derivative of for :

Theorem (Derivative of power functions)

The power function

is for differentiable on all of . For alle there is

Proof (Derivative of power functions)

For , so there is

We used the geometric sum formula and the continuity of the polynomial function .

Polynomials and rational functions[Bearbeiten]

Using the calculation rules for derivatives we can now calculate the derivatives of polynomial functions and rational functions:

Theorem (Derivative of polynomial functions)

Let

with and be a polynomial function of degree . Then is differentiable on all of , and for there is

Proof (Derivative of polynomial functions)

Using the derivative rule for the multiple of a function , every single summand of the polynomial is differentiable on . With the summation rule we can derive every polynomial function term by term on and obtain for :

with the derivative of the zeroth summand disappearing.

In particular, it follows for and that linear and quadratic functions are differentiable onn all of .

Exercise (Derivative of rational functions)

Let

with and a rational function defined on . Show that is differentiable on , and calculate the derivative.

Solution (Derivative of rational functions)

Numerator and denominator of are polynomials. Since the denominator is non-zero on and polynomials are differentiable, it follows from the quotient rule that is differentiable on .

Further there is for :

Power functions with integer powers[Bearbeiten]

We can already differentiate power functions with natural powers. Now we investigate those with negative integer exponents.

Example (Derivative of the hyperbolic function)

The power function

is differentiable on and there is

for .

Exercise (Derivative of )

Prove that the power function

is differentiable on and compute its derivative.

Solution (Derivative of )

For there is

In the general case with there is

Theorem (Derivative of the power function with negative integer powers)

The power function

is differentiable on , and for there is

Proof (Derivative of the power function with negative integer powers)

For there is

Exercise (Derivative of the power function)

Prove using the quotient rule

Solution (Derivative of the power function)

For there is by the quotient rule

Remark: Of course we can also apply the inverse rule directly, and thus get the same result

Let us look again at the derivatives rule in the last case, i.e. for . If we put , we get . The derivative rule is hence the same as for with . So we can summarize the two cases and get

Theorem (Derivative of the power function with natural powers)

For the power function

is differentiable on . For there is then

In the case of it is even differentiable on all of .

Root functions[Bearbeiten]

Now we investigate the derivative of root functions. We start again with the simplest case:

Example (Derivative of the square root function)

The square root function

is differentiable on and for there is

Question: Why is the square root function in not differentiable, although it is defined and continuous there?

For the differential quotient there is

So it does not exist. Hence, we have non-differentiability.

Exercise (Derivative of the cubic root function)

Compute the derivative of the cubic root function

Solution (Derivative of the cubic root function)

For there is

Now let us consider the general case of the -th root function. Here there is

Theorem (Derivative of the -th root function)

Let . Then the -th root function

is differentiable on , and for there is

Proof (Derivative of the -th root function)

For there is

This can now be generalised

Theorem (Derivative of the generalized root function)

For and , the generalized root function

is differentiable on , and for there is

Proof (Derivative of the generalized root function)

Since on die functions and are differentiable, the chain rule implies at that

Hint

For and and the power fucniton with rational exponent was defined as

So for we also have the derivative rule

The (generalized) exponential function and generalized power functions[Bearbeiten]

In this section we prove that the derivative of the exponential function is again the exponential function. So we can determine the derivative of the generalized exponential and power function.

Theorem (Derivative of the exponential function)

The exponential function

is differentiable on , and for there is

How to get to the proof? (Derivative of the exponential function)

For this derivative it is more useful to use the method

Because in this case we know the limit value

Furthermore we need the functional equation of the exponential function

Proof (Derivative of the exponential function)

For there is

Using the chain rule, the derivatives of the generalized exponential function for and the generalized power function for can be calculated:

Theorem (Derivative of the generalized exponential function)

For the generalized exponential function

is differentiable on , and for there is

Proof (Derivative of the generalized exponential function)

For there is

Theorem (Derivative of the generalized exponential function)

For the generalized exponential function

is differentiable on , and for there is

Exercise (Derivative of the generalized exponential function)

Prove that the derivative of the generalized power function at is .

Proof (Derivative of the generalized exponential function)

For the chain rule yields

Logarithmic functions[Bearbeiten]

Now we turn to the derivative of the natural and generalised logarithm function. Since the natural logarithm is the inverse of the exponential function, we can deduce its derivative directly from rule for derivatives of inverse function:

Theorem (Derivative of the natural logarithm function)

The natural logarithm function

is differentiable on . For there is

Proof (Derivative of the natural logarithm function)

For the exponential function there is: . So the function is differentiable, and because of strictly monotonously increasing. Furthermore, is surjective. The inverse function is the (natural) logarithm function

From the theorem about the derivative of the inverse function we now have for every :

The derivative can also be calculated directly using the differential quotient. If you want to try this, we recommend the corresponding exercise (missing).

Using the derivative of the natural logarithm function we can now immediately conclude

Theorem (Derivative of the generalized logarithm function)

For the generalized logarithm function

is differentiable on . For there is

Proof (Derivative of the generalized logarithm function)

From the derivative rule for the multiple of a function, we get that for all :

If the derivative of the natural logarithm is not available, we can calculate it using the theorem of the derivative of the inverse function.

Trigonometric functions[Bearbeiten]

Sine[Bearbeiten]

Theorem (Derivative of the sine function)

The sine function is differentiable. For all there is:

Proof (Derivative of the sine function)

For there is

Cosine[Bearbeiten]

Theorem (Derivative of the cosine function)

The cosine function is differentiable with

Proof (Derivative of the cosine function)

Tangent[Bearbeiten]

Theorem (Derivative of the tangent function)

The tangent function

is differentiable on , and for there is

Proof (Derivative of the tangent function)

Since for , the function is differentiable by the quotient rule, and for there is

Exercise (Derivative of the cotangent function)

The cotangent function

is differentiable on , and for there is

Solution (Derivative of the cotangent function)

Since for , the function is differentiable by the quotient rule, and for there is

Alternative solution:

The derivatives of secant and cosecant can be found in the corresponding exercise.

arc-functions[Bearbeiten]

Using the rule for derivatives of the inverse function we can differentiate the arc-functions (which are inverses of sine, cosine, etc.)

arcsin and arccos[Bearbeiten]

Theorem (Derivative of the arcsin/arccos function)

The inverse functions of the trigonometric functions , are differentiable with

Note: and are defined and continuous on , but only differentiable on .

Proof (Derivative of the arcsin/arccos function)

Derivative of :

For the sine function there is: . So the function is differentiable, and since for all , it is strictly monotonously increasing on this interval. Further, . So is surjective. The inverse function is the arc sine function

From the theorem about the derivative of the inverse we now have for every :

Derivative of :

For the cosine function there is: . So the function is differentiable, and because of , strictly monotonously decreasing. Further, . So is surjective. The inverse function

is differentiable according to the theorem about the derivative of the inverse function, and for every there is:

arctan and arccot[Bearbeiten]

Theorem (Derivative of the arctan/ arccot function)

The inverse functions of the trigonometric functions , are differentiable, and there is

Proof (Derivative of the arctan/ arccot function)

For the tangent function there is: . So the function is differentiable and strictly monotonically increasing. Further, . So is surjective. The inverse function

is hence differentiable, and now for there is:

Hyperbolic functions[Bearbeiten]

And finally, we determine the derivatives of the hyperbolic functions , and :

Theorem (Derivative of hyperbolic functions)

The functions

are differentiable, and there is

Proof (Derivative of hyperbolic functions)

The derivatives follow directly from the calculation rules. We show only the derivative of . The other two are left to you for practice.

According to the factor and difference rule for all is differentiable, and there is

Exercise (Derivative of and )

Prove that and are differentiable with

and

Proof (Derivative of and )

Derivative of :

According to the factor and sum rule, is differentiable for all , and there is

Derivative of :

is differentiable on all of by the quotient rule and there is