Category: Derivative

Definition of Derivative

There is one limit that is used very frequently in applications of calculus and different sciences. This limit has form `lim_(h->0)(f(x+h)-f(x))/h` and has special notation.

Definition. The derivative of function `f` is a function that is denoted by `f'(x)` and calculated as `f'(x)=lim_(h->0)(f(x+h)-f(x))/h`.

Derivatives of Elementary Functions

Let's start with the simplest function, namely, the constant polynomial `f(x)=c`.

Derivative of a Constant Function `d/(dx)(c)=0`.

Indeed, `f'(x)=lim_(h->0)(f(x+h)-f(x))/h=lim_(h->0)(c-c)/h=lim_(h->0)0=0`.

Table of the Derivatives

Below is the list of the most common derivatives.

`f(x)` `f'(x)` Power Rule `x^n` `nx^(n-1)` Exponential Function `a^x` `ln(a)a^x` `e^x` `e^x` Logarithmic Function `log_a(x)` `1/(xln(a))` `ln|x|` `1/x` Trigonometric Functions `sin(x)` `cos(x)` `cos(x)` `-sin(x)` `tan(x)` `1/(cos^2(x))=sec^2(x)` `cot(x)` `-1/(sin^2(x))=-csc^2(x)` `sec(x)=1/(cos(x))` `sec(x)tan(x)` `csc(x)=1/(sin(x))` -`csc(x)cot(x)` Inverse Trigonometric Functions `arcsin(x)` `1/(sqrt(1-x^2))` `arccos(x)` `-1/(sqrt(1-x^2))` `arctan(x)` `1/(1+x^2)` `text(arccot)(x)` `-1/(1+x^2)` `text(arcsec)(x)` `1/(xsqrt(x^2-1))` `text(arccsc)(x)` `-1/(xsqrt(x^2-1))` Hyperbolic Functions `sinh(x)` `cosh(x)` `cosh(x)` `sinh(x)` `tanh(x)` `1/(cosh^2(x))=text(sech)^2(x)` `coth(x)` `-1/(sinh^2(x))=-text(csch)^2(x)` `text(sech)(x)=1/(cosh(x))` `-text(sech)(x)tanh(x)` `text(csch)(x)=1/(sinh(x))` `-text(csch)(x)coth(x)` Inverse Hyperbolic Functions `text(arcsinh)(x)` `1/(sqrt(x^2+1))` `text(arccosh)(x)` `1/(sqrt(x^2-1))` `text(arctanh)(x)` `1/(1-x^2)` `text(arccot)text(h)(x)` `1/(1-x^2)` `text(arcsec)text(h)(x)` `-1/(xsqrt(1-x^2))` `text(arccsc)text(h)(x)` `-1/(|x|sqrt(1+x^2))` Differentiation Rules `c` `0` `g(x)+h(x)` `g'(x)+h'(x)` `g(x)-h(x)` `g'(x)-h'(x)` `c*g(x)` `c*g'(x)` `g(x)h(x)` `g'(x)h(x)+g(x)h'(x)` `(g(x))/(h(x))` `(g'(x)h(x)-g(x)h'(x))/(h^2(x))` `g(h(x))` `g'(h(x))*h'(x)` `f^(-1)(x)` `1/(f'(f^(-1)(x)))`

Tangent Line, Velocity and Other Rates of Changes

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

The Tangent Line

Suppose we are given curve `y=f(x)` and point on curve `P(a,f(a))`. We want to find equation of tangent line at point `P`.

Studying Derivative Graphically

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

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