![]() Which, of course, is exactly the same result that you would obtain with implicit differentiation. This is the chain rule.īecause the the demand equation consists of the sum of two smaller expressions, the derivative sum rule says that we can simply add the derivatives of each expression. For problems 1 6 use the Product Rule or the Quotient Rule to find the derivative of the given function. For each problem, you are given a table containing some values of differentiable functions f (x). To implicitly differentiate the demand equation with respect to time, you first differentiate each variable with respect to itself, and then multiply by the derivative of that variable with respect to time. Implicit Differentiation with respect to time If y is written in terms of x, i.e., yf(x), then this is easy to do using the chain rule: ydydtdydxdxdtdydxx. In words, the chain rule says the derivative of a composite function is the derivative of the outside function, done to the inside function, times the. 0:00 / 9:20 Word problem using related rates and chain rule Midnighttutor 11.4K subscribers Subscribe Save 20K views 16 years ago See for all the latest calculus. You can calculate the value of $x$ and $y$ at time = 5 using those functions, and you can calculate the values of $dx \over dt$ and $dy \over dt$ by taking the derivative of those functions. You are also explicitly given information on how $x$ and $y$ are functions of time. ![]() The former approach allows you to break it into "smaller pieces", though the end result is the same. I find that the latter approach gets a little confusing here, so, in this instance, I prefer the former. ![]() ![]() You find the derivative either by using partial derivatives, as suggested by or you can implicitly differentiate with respect to time. It is a bit more challenging here because of the fractional exponents. Be able to compare your answer with the direct method of computing the partial derivatives. You need to differentiate the demand equation. ![]()
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