Category: Derivative

Definition of Derivative

There is one limit that is used very frequently in the applications of calculus and different sciences. This limit has the form $$$\lim_{{{h}\to{0}}}\frac{{{f{{\left({x}+{h}\right)}}}-{f{{\left({x}\right)}}}}}{{h}}$$$ and has a special notation.

Derivatives of Elementary Functions

Let's start with the simplest function, namely the constant polynomial $$${f{{\left({x}\right)}}}={c}$$$.

Derivative of a Constant Function. $$$\frac{{d}}{{{d}{x}}}{\left({c}\right)}={0}$$$.

Indeed, $$${f{'}}{\left({x}\right)}=\lim_{{{h}\to{0}}}\frac{{{f{{\left({x}+{h}\right)}}}-{f{{\left({x}\right)}}}}}{{h}}=\lim_{{{h}\to{0}}}\frac{{{c}-{c}}}{{h}}=\lim_{{{h}\to{0}}}{0}={0}$$$.

Table of Derivatives

Below is the list of the most common derivatives.

$$$f{{\left({x}\right)}}$$$ $$$f{'}\left({x}\right)$$$ Power Rule $$$x^n$$$ $$$nx^{n-1}$$$ Exponential Function $$$a^x$$$ $$${\ln{{\left({a}\right)}}}{{a}}^{{x}}$$$ $$${{e}}^{{x}}$$$ $$${{e}}^{{x}}$$$ Logarithmic Function $$${\log}_{{a}}{\left({x}\right)}$$$ $$$\frac{{1}}{{{x}{\ln{{\left({a}\right)}}}}}$$$ $$${\ln}{\left|{x}\right|}$$$ $$$\frac{{1}}{{x}}$$$ Trigonometric Functions $$${\sin{{\left({x}\right)}}}$$$ $$${\cos{{\left({x}\right)}}}$$$ $$${\cos{{\left({x}\right)}}}$$$ $$$-{\sin{{\left({x}\right)}}}$$$ $$${\tan{{\left({x}\right)}}}$$$ $$$\frac{{1}}{{{{\cos}}^{{2}}{\left({x}\right)}}}={{\sec}}^{{2}}{\left({x}\right)}$$$ $$${\cot{{\left({x}\right)}}}$$$ $$$-\frac{{1}}{{{{\sin}}^{{2}}{\left({x}\right)}}}=-{{\csc}}^{{2}}{\left({x}\right)}$$$ $$${\sec{{\left({x}\right)}}}=\frac{{1}}{{{\cos{{\left({x}\right)}}}}}$$$ $$${\sec{{\left({x}\right)}}}{\tan{{\left({x}\right)}}}$$$ $$${\csc{{\left({x}\right)}}}=\frac{{1}}{{{\sin{{\left({x}\right)}}}}}$$$ $$$-{\csc{{\left({x}\right)}}}{\cot{{\left({x}\right)}}}$$$ Inverse Trigonometric Functions $$${\operatorname{arcsin}{{\left({x}\right)}}}$$$ $$$\frac{{1}}{{\sqrt{{{1}-{{x}}^{{2}}}}}}$$$ $$${\operatorname{arccos}{{\left({x}\right)}}}$$$ $$$-\frac{{1}}{{\sqrt{{{1}-{{x}}^{{2}}}}}}$$$ $$${\operatorname{arctan}{{\left({x}\right)}}}$$$ $$$\frac{{1}}{{{1}+{{x}}^{{2}}}}$$$ $$$\text{arccot}{\left({x}\right)}$$$ $$$-\frac{{1}}{{{1}+{{x}}^{{2}}}}$$$ $$$\text{arcsec}{\left({x}\right)}$$$ $$$\frac{{1}}{{{x}\sqrt{{{{x}}^{{2}}-{1}}}}}$$$ $$$\text{arccsc}{\left({x}\right)}$$$ $$$-\frac{{1}}{{{x}\sqrt{{{{x}}^{{2}}-{1}}}}}$$$ Hyperbolic Functions $$${\sinh{{\left({x}\right)}}}$$$ $$${\cosh{{\left({x}\right)}}}$$$ $$${\cosh{{\left({x}\right)}}}$$$ $$${\sinh{{\left({x}\right)}}}$$$ $$${\tanh{{\left({x}\right)}}}$$$ $$$\frac{{1}}{{{{\cosh}}^{{2}}{\left({x}\right)}}}={\text{sech}}^{{2}}{\left({x}\right)}$$$ $$${\coth{{\left({x}\right)}}}$$$ $$$-\frac{{1}}{{{{\sinh}}^{{2}}{\left({x}\right)}}}=-{\operatorname{csch}}^{{2}}{\left({x}\right)}$$$ $$$\text{sech}{\left({x}\right)}=\frac{{1}}{{{\cosh{{\left({x}\right)}}}}}$$$ $$$-\text{sech}{\left({x}\right)}{\tanh{{\left({x}\right)}}}$$$ $$$\operatorname{csch}{\left({x}\right)}=\frac{{1}}{{{\sinh{{\left({x}\right)}}}}}$$$ $$$-\operatorname{csch}{\left({x}\right)}{\coth{{\left({x}\right)}}}$$$ Inverse Hyperbolic Functions $$$\text{arcsinh}{\left({x}\right)}$$$ $$$\frac{{1}}{{\sqrt{{{{x}}^{{2}}+{1}}}}}$$$ $$$\text{arccosh}{\left({x}\right)}$$$ $$$\frac{{1}}{{\sqrt{{{{x}}^{{2}}-{1}}}}}$$$ $$$\text{arctanh}{\left({x}\right)}$$$ $$$\frac{{1}}{{{1}-{{x}}^{{2}}}}$$$ $$$\text{arccot}\text{h}{\left({x}\right)}$$$ $$$\frac{{1}}{{{1}-{{x}}^{{2}}}}$$$ $$$\text{arcsec}\text{h}{\left({x}\right)}$$$ $$$-\frac{{1}}{{{x}\sqrt{{{1}-{{x}}^{{2}}}}}}$$$ $$$\text{arccsc}\text{h}{\left({x}\right)}$$$ $$$-\frac{{1}}{{{\left|{x}\right|}\sqrt{{{1}+{{x}}^{{2}}}}}}$$$ Differentiation Rules $$${c}$$$ $$${0}$$$ $$${g{{\left({x}\right)}}}+{h}{\left({x}\right)}$$$ $$${g{'}}{\left({x}\right)}+{h}'{\left({x}\right)}$$$ $$${g{{\left({x}\right)}}}-{h}{\left({x}\right)}$$$ $$${g{'}}{\left({x}\right)}-{h}'{\left({x}\right)}$$$ $$${c}\cdot{g{{\left({x}\right)}}}$$$ $$${c}\cdot{g{'}}{\left({x}\right)}$$$ $$${g{{\left({x}\right)}}}{h}{\left({x}\right)}$$$ $$${g{'}}{\left({x}\right)}{h}{\left({x}\right)}+{g{{\left({x}\right)}}}{h}'{\left({x}\right)}$$$ $$$\frac{{{g{{\left({x}\right)}}}}}{{{h}{\left({x}\right)}}}$$$ $$$\frac{{{g{'}}{\left({x}\right)}{h}{\left({x}\right)}-{g{{\left({x}\right)}}}{h}'{\left({x}\right)}}}{{{{h}}^{{2}}{\left({x}\right)}}}$$$ $$${g{{\left({h}{\left({x}\right)}\right)}}}$$$ $$${g{'}}{\left({h}{\left({x}\right)}\right)}\cdot{h}'{\left({x}\right)}$$$ $$${{f}}^{{-{1}}}{\left({x}\right)}$$$ $$$\frac{{1}}{{{f{'}}{\left({{f}}^{{-{1}}}{\left({x}\right)}\right)}}}$$$

Tangent Line, Velocity, and Other Rates of Changes

Now we are going to talk about the problems that lead to the concept of derivative.

The Tangent Line

Suppose that we are given a curve $$$y={f{{\left({x}\right)}}}$$$ and a point on the curve $$${P}{\left({a},{f{{\left({a}\right)}}}\right)}$$$. We want to find the equation of the tangent line at the point $$$P$$$.

Studying Derivative Graphically

In this note, we are going to talk about how to sketch the graph of the derivative knowing the graph of the function and see graphically when the function is not differentiable.

Example. The graph of the function is shown. Sketch the graph of the derivative.