HallsofIvy said:No, it isn't.
SteamKing said:What happens to the exponent when we differentiate?
SteamKing said:That looks better.
It's much simpler to write the equation in this form:musicgold said:Oh! How about this?
dI/dS = (- N U/ D2 ) / ( S U/ D + 1)2
Mark44 said:Also, to find the sensitivity of I to small changes in S, write the equation above using differentials.
dI = ##\frac{-NU}{(US + D)^2} dS ##
For small changes in S, you can approximate dS by ΔS and dI by ΔI.
ModestyKing said:What Mark is saying (I believe) is that if you replace dI with change in I (the answer you want to get) and dS with change in S (which you have; 1% = 0.01), then you're set. The concept is this: dI/dS is the change of dI over dS; the change of output compared to input. So, in dI/dS, set dS to what it is; the change in input, or 0.01. Now multiply both sides by 0.01 to get dI = (rest of equation). There, you've got a ratio! ^_^
Mark44 said:You didn't show what you did to get your value of dI, so it's hard to say what you're missing. Using the formula I wrote in post #8, I get dI ≈ -8.93 X 10-5.
Differentiation is a mathematical process used to find the rate at which one variable changes with respect to another variable. It is a fundamental concept in calculus and is used to solve problems involving rates of change, optimization, and related rates.
Differentiation is done by taking the derivative of a function. This involves finding the slope of the function at a specific point by calculating the limit of the function as the change in the independent variable approaches zero. The resulting derivative represents the instantaneous rate of change of the function at that point.
The purpose of differentiation is to analyze and understand the behavior of functions. It allows us to find the maximum and minimum values of a function, determine the direction in which the function is increasing or decreasing, and solve optimization problems. Differentiation is also used in physics, engineering, and other fields to model and solve real-world problems.
In theory, differentiation is always correct as it follows a set of rules and principles. However, it is possible to make mistakes when applying these rules, resulting in an incorrect derivative. It is important to double-check the work and use different methods, such as graphing, to confirm the result.
Differentiation can be used for most functions, including polynomial, exponential, logarithmic, and trigonometric functions. However, there are some functions that are not differentiable, such as discontinuous or non-continuous functions. In these cases, other methods, such as numerical approximation, may be used.