CompuChip said:It is basically just application of the chain rule.
Calling y = h(u),
[tex]\frac{dF(y)}{du} = \frac{dF(y)}{dy} \times \frac{dy}{du}[/tex]
so if you call dF(y)/dy = f(y) then you have your identity (just replace y with its definition h(u) again).
kingwinner said:OK, thanks!
But can we replace y by h(u) and vice versa that freely?
Sorry my calculus is a bit rusty now...
Differentiation is a mathematical process that involves finding the rate of change of a function, or its slope, at a specific point. It is a fundamental concept in calculus and is used to solve problems involving rates of change in various fields such as physics, engineering, and economics.
While differentiation is the process of finding the slope of a function, integration is the reverse process of finding the function itself from its slope. In other words, differentiation deals with the instantaneous rate of change, while integration deals with the cumulative change over a given interval.
Differentiation has various applications in real-life problems, such as finding the maximum or minimum value of a function, calculating the velocity and acceleration of moving objects, determining the rate of change in financial markets, and optimizing production processes.
A composite function is a function that is formed by combining two or more functions. It is written as f(g(x)), where the output of one function, g(x), becomes the input of another function, f(x). This allows us to represent complicated relationships between variables in a more simplified form.
To differentiate a composite function, we use the chain rule, which states that the derivative of a composite function is equal to the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. In other words, we take the derivative of the outer function, leaving the inner function unchanged, and then multiply it by the derivative of the inner function.