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I don't knowSimon Bridge said:
yes, derivative is the rate of one function to another function, it actually says how fast one function changes with respect to other, am I right?Simon Bridge said:
Not exactly ... that was the description of what the derivative is, not the definition. The definition is: $$f^\prime(x) = \lim_{\Delta x\to 0}\frac{f(x+\Delta x) - f(x)}{\Delta x}$$ ... this gives the derivative of f(x) with respect to x.
got it. Thank youSimon Bridge said:
The product rule is a formula used in calculus to find the derivative of a product of two functions. It states that the derivative of the product of two functions is equal to the first function multiplied by the derivative of the second function, plus the second function multiplied by the derivative of the first function.
The product rule should be used when taking the derivative of a function that is a product of two or more functions. It cannot be used for functions that are sums, differences, or quotients.
To apply the product rule, first identify the two functions that are being multiplied together. Then, take the derivative of each function separately. Finally, plug these derivatives into the product rule formula and simplify the resulting expression.
Yes, the product rule can be extended to any number of functions being multiplied together. The general formula is the derivative of the first function multiplied by the product of the remaining functions, plus the derivative of the second function multiplied by the product of the remaining functions, and so on.
The product rule allows us to find the derivative of a product of functions without having to use the limit definition of the derivative. This makes it a more efficient and convenient method for finding derivatives of more complex functions. It is also a fundamental rule in calculus and is used in many applications, such as optimization problems and curve sketching.
