Skip to article frontmatterSkip to article content

6. Derivative of a function

Faculty of Economics and Business
University of Zagreb

6.1Introduction

Visual interpretation of the derivative - the derivative of a function ff at a point xx is equal to the slope of the tangent line to the graph of the function ff at a point xx.

6.2Derivative of elementary functions

Basic rules for differentiation
(f+g)′=f′+g′(c⋅f)′=c⋅f′(f⋅g)′=f′⋅g+f⋅g′(fg)′=f′⋅g−f⋅g′g2\begin{align*} (f+g)' &= f' + g' & & & & & (c \cdot f)' &= c \cdot f' \\ (f \cdot g)' &= f' \cdot g + f \cdot g' & & & & & \left( \frac{f}{g}\right)' &= \frac{f' \cdot g - f \cdot g'}{g^2} \\ \end{align*}
(1)
Table of derivatives
c′=0(xn)′=n⋅xn−1(x)′=12x(ax)′=ax⋅ln⁡(a)(ex)′=ex(ln⁡(x))′=1x\begin{align*} c' &= 0 & & & & & (x^n)' &= n \cdot x^{n-1} & & & & & (\sqrt{x})' &= \frac{1}{2\sqrt{x}}\\ (a^x)' &= a^x \cdot \ln(a) & & & & & (e^x)' &= e^x & & & & & (\ln(x))' &= \frac{1}{x} \end{align*}
(2)

6.3Derivative of composition of functions

The following formula tells us how to find derivative of compositions:

Therefore, in order to find the derivative of a composition of functions you move from the “outside” to the “inside” and write the derivatives as you go along.

6.4Derivative via the definition

6.5Higher-order derivatives

Part 2. Functions of one variable
5. Domains and limits of functions
Part 2. Functions of one variable
7. Differential of a function