Derivation rules of functions

The knowledge of derivation rules of functions is important for the application of mathematics in all sciences. In our case, the application of derivation rules plays a major role in the interpretation of economic and business models. In this article, we will therefore introduce you to some of the basic derivation rules that you should master for your analysis in your business studies.

Derivation rules of functions in mathematics – differential calculus

Derivation rules apply to several basic functions in mathematics and consequently, specify how we can exploit the general form of mathematical functions to find the functional form of the derivative of a function. Since this topic is not new to science, you will find the general form of the derivation rules in most mathematical literature. In this article, we will introduce the following derivation rules for functions with an independent function:

• Constant rule
• Power rule
• Factor rule
• Sum rule
• Product rule
• Ratio rule
• Chain rule
• Rules for exponential functions
• Rules for logarithmic functions

So why do you need to understand how to calculate the derivative of a function?

The derivative of a function explains the infinitesimal change of the dependant variable of a function when the change of the independent variable converges to zero. In this case, we are looking for the marginal change of the dependent variable, when the dependent variable increases by one unit.

How do you get from secant slope to tangent slope?

First, formulate a mathematical statement for the average slope of a function between two coordinate points. This is exactly the secant slope of a function between two points of a function.

\frac {\Delta f(x)}{\Delta x} = \frac{f(x_0+\Delta x)-f(x_0)}{\Delta x}

Second, define the limit theorem for the secant slope to derive the derivation rule of general functions of any type. The limit theorem of the secant slope describes the tag slope of the function at the point $x_0$

f'(x_0)= \lim \limits_{ \Delta x \to 0} \frac {\Delta f(x)}{\Delta x} =  \lim \limits_{ \Delta x \to 0}  \frac{f(x_0+\Delta x)-f(x_0)}{\Delta x}

Constant rule for constant function $f(x)=C$

First, the constant rule for constant functions $f(x)=C$ with $C$ as constant is:

f'(x)=0

Examples of constant functions in economics and business administration are fixed cost functions. Such fixed cost functions describe costs $K_f(x)=C$ that do not depend on the level of production. By applying derivative conditions to constant functions, economists consequently argue that fixed costs do not directly affect our marginal cost decision $K'(x)=K’_v(x)$. We will learn that the average function change must also be considered in cost analysis.

Power rule for power function $f(x)=x^n$

Secondly, the power rule for the power function $f(x)=x^n$ with $n$ as power is:

f'(x)=nx^{(n-1)}

Factor rule for general functions $f(x)=C \cdot g(x)$

Thirdly, the factor rule for general functions $f(x)=C \cdot g(x)$ with $C$ as constant is:

f'(x)=C \cdot g'(x)

Summation rule for general functions $f(x)=g(x) \pm h(x)$

The summation rule for general functions $f(x)=g(x) \pm h(x)$ is applied to the derivative of the addition and substrate of two summands as follows:

f'(x)=g'(x) \pm h'(x)

Product rule for general functions $f(x)=u(x) \cdot v(x)$

The product rule of the derivative for general functions $f(x)=u(x) \cdot v(x)$ is applied when a multiplicative form of the function is recognizable and reads:

f'(x)=u'(x) \cdot v(x)+u(x) \cdot v'(x)

Ratio rule for general functions $f(x)=\frac{u(x)} {v(x)}$

Ratio rule for general functions $f(x)=\frac{u(x)} {v(x)}$ is applied when a quotient or fraction is recognizable in the function form and reads:

f'(x)=\frac{u'(x) \cdot v(x)-u(x) \cdot v'(x)} {v(x)^2}

Chain rule for general functions $f(x)=f(g(x))$

We can apply the chain rule for general functions $f(x)=f(g(x))$ when a chaining sequence of the functional form is recognizable and reads:

f'(x)=f'_g \cdot g'(x))  \text {with} \ f'_g \ \text{as the derivative of the function} \ f(g) \ \text{to} \ g

Derivation rule of the exponential function $f(x)=e^x$

Derivation rule of the exponential function $f(x)=e^x$ is:

f'(x)=e^x

Be careful when using the derivative rule for the exponential function, because the derivative rule is sensitive to the positivity of the power. The correct way to derive the exponential function is to always use the chain rule of the derivative.

Derivation rule of the logarithmic function $f(x)=\ln(x)$

The first derivation rule of the logarithm function $f(x)=\ln(x)$ is:

f'(x)=\frac{1}{x}

You can also read this article about derivation rules for functions in German.