can anybody tell me how the Leibniz's rule for the differentiation work?
I have not studied partial differentiation yet...
This is better known as the product rule.
http://en.wikipedia.org/wiki/Product_rule
Take f(x)=x^2 sin x for instance.
f'=sin x d/dx x^2 + x^2 d/dx sinx
There are several different rules for differentiation attributed to Leibniz. One os sexplained at
http://www.karlscalculus.org/calc8_4.html
Another has to do with differentiation of a definite integral, with respect to a variable that appears in the integrand and/or the limits of integeration.
Of course! I can explain Leibniz's rule for differentiation to you.
Leibniz's rule, also known as the product rule for differentiation, allows us to find the derivative of the product of two functions. It is used when we have a function that is the product of two other functions, and we want to find its derivative.
The product rule states that if you have two functions, let's call them f(x) and g(x), and their product h(x) = f(x) * g(x), then the derivative of h(x) with respect to x (denoted as h'(x) or dy/dx) can be found using the following formula:
h'(x) = f'(x) * g(x) + f(x) * g'(x)
Let's break down the formula to understand how it works. The first term f'(x) * g(x) represents the derivative of the first function (f(x)) multiplied by the second function (g(x)). It captures how the first function's rate of change affects the overall rate of change.
The second term f(x) * g'(x) represents the first function (f(x)) multiplied by the derivative of the second function (g(x)). It captures how the second function's rate of change affects the overall rate of change.
By summing these two terms, we obtain the derivative of the product of the two functions.
To apply Leibniz's rule, you need to know the derivatives of the individual functions, f(x) and g(x). If you are not familiar with partial differentiation yet, it means that you may need to know the general rules of differentiation for various types of functions, such as power functions, exponential functions, trigonometric functions, etc. These rules can be found in a calculus textbook or online resources.
Once you have the derivatives of f(x) and g(x), substitute them into the formula h'(x) = f'(x) * g(x) + f(x) * g'(x) to find the derivative of the product function, h(x).
If you haven't covered partial differentiation yet, I recommend starting with basic differentiation rules and concepts before diving into Leibniz's rule for the product of functions.