![]() ![]() The derivative of the product of these two functions uv is given by: Fortunately, there is a simple formula (the product rule) that we can use to find the derivative of the product of two functions.Įssentially, the rule states that in order to find the derivative of the product of two functions, we take the first function multiplied by the derivative of the second function, and add it to the second function multiplied by the derivative of the first function. There will be occasions, however, when this will much more difficult or impossible to do. ![]() In this case, multiplying out the two functions was a relatively trivial exercise, and we were able to find the derivative of the function ƒ( x) = x( x 2 + 1) without difficulty. This is obviously very different from the result we got by simply multiplying together the derivatives of the functions ƒ(x) = x and ƒ( x) = x 2 + 1. Taking the derivative of x 3 + x, we get: Multiplying out the brackets in our original function, however, we get: Obviously, if we multiply these two derivatives together, the result will be x. We'll start by finding the derivative of each function separately: This function is the product of two functions, ƒ( x) = x and ƒ( x) = x 2 + 1. Suppose we need to find the derivative of the function ƒ( x) = x( x 2 + 1). An example should serve to illustrate the point. Unfortunately, it is not quite that simple. two functions multiplied together), we can simply multiply together the derivatives of each function. ![]() You might therefore be tempted to assume that, for a function that is the product of two functions (i.e. We know that we can find the differential of a polynomial function by adding together the differentials of the individual terms of the polynomial, each of which can be considered a function in its own right. This is very easy to do by replacing different occurrences of $x$ with separate variables, computing the partial derivatives, adding them up and setting all the variables to the same value $x$.The product rule gives us the derivative of the product of two (or more) functions. To understand why the above technique is useful try to compute the derivative of functions such as $f(x)=x^x$. For me a part of being intuitive is the ability of immediately detect pattern and use it in other circumstances, and this approach goes well beyond the product rule. $d(g\cdot h)(x,x) = h\cdot g'(x) \cdot dx + g\cdot h'(x) \cdot dx$ĭifferent people have different notions about what is intuitive. $d(g\cdot h)(y,z) = h(z)\cdot g'(y) \cdot dy + g(y)\cdot h'(z) \cdot dz$įinally, it remains to consider what happens when both $y$ and $z$ have the same value $x$: Because $g(y)$ is constant with respect to $z$ and $h(z)$ is constant with respect to $y$ and differentiation is linear we have: Now suppose that $f$ splits into a product of two functions, each being a function of just one of the variables: $f=g(y)\cdot h(z)$. The above is just a generalization of the chain rule, and IMO is very intuitive. $df=\partial f/\partial y \cdot dy + \partial f/\partial z \cdot dz$ Specifically for the product rule, take a function of two variables $f(y,z)$ and consider the formula for the differential of $f$: For me intuition for product rule, as well as a couple of other techniques, comes from multi-variable calculus. ![]()
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