Several ways for finding the derivative of a function
As we know, derivatives are all about change. It is the rate of change of a quantity as compared to the another one according to the derivative of a function definition. Just like velocity is the rate of change of distance with respect to the change in time. Let us consider that y is a function of x, i.e. y = f(x) then dy/dx is called the derivative of a function. According to the definition of the derivative of a function, it gives the rate of change of y with respect to x.
Calculus limits provide basis for finding derivatives of a function. The derivative of function y = f(x) at x = a can be given as if the limits exist. It can be denoted by f' (a). Now consider f(x) for a closed interval [a,b]. Then the rate of change can be given byFor the linear functions, this concept holds true for linear functions. But for non-linear functions, it just provides the average rate. Consider a infinitely small point h that is close to value at f(x). Now finding the derivative of a function, it is given by the formula
It's easy graphing the derivative of a function
Now we will be graphing the derivative of a function for y = f(x) = ln(x) which comes 1/x as shown below.
Some important derivative function examples
Some of the important derivative function examples are given below to get a quick overview of derivatives.
(1) Find the derivative of x2:
Here f(x) = x2. Hence f(x + h) = (x + h)2 = x2 + 2xh + h2. Taking the difference, f(x + h) – f(x) = h2 + 2xh. Thus f'(x) = [(h2 + 2xh) / h]h->0 = [h + 2x]h->0 = 2x. Thus derivative of x2 comes to be 2x.
(2) Find the f'(x) if f(x) = 3x:
Here f(x) = 3x. Hence f(x + h) = 3x + 3h which yields f(x + h) - f(x) = 3x + 3h – 3x = 3h. Now taking the ratio, f'(x) = [(f(x + h) – f(x)) / h]h->0 = [3h / h]h->0 = 3. Thus derivative of 3x is a constant value 3.
San Francisco, USA
"If you're at school or you just deal with mathematics, you need to use Studygeek.org. This thing is really helpful."
"I will recommend Studygeek to students, who have some chalenges in mathematics. This Site has bunch of great lessons and examples. "
" I'm a geek, and I love this website. It really helped me during my math classes. Check it out) "
"I use Studygeek.org a lot on a daily basis, helping my son with his geometry classes. Also, it has very cool math solver, which makes study process pretty fun"