PedroB said:Homework Statement
Find the derivative of the function f(x)=cos(√x) by first principles
Homework Equations
f'(x)= lim as h tends to zero of [f(x+h)-f(x)]/h
The Attempt at a Solution
Problems arise immediately, since I have no idea what to do with the expression cos(√(x+h)), I've tried binomial expansion, but to no avail. Any help would be greatly appreciated
Derivation by first principles is a mathematical method used to find the derivative of a function by using the definition of a derivative. It involves taking the limit of the difference quotient as the change in the input variable approaches zero.
Derivation by first principles allows us to find the exact rate of change of a function at a specific point, without relying on pre-existing knowledge or formulas. It is the most fundamental method for finding derivatives and forms the basis for more advanced techniques.
To find the derivative of cos(x^0.5) using first principles, we first write out the definition of a derivative and substitute our function in for f(x). Then, we simplify the difference quotient as much as possible and take the limit as the change in x approaches zero. This will give us the derivative of cos(x^0.5) in terms of x.
Derivation by first principles allows for a more precise and accurate calculation of derivatives, as it does not rely on pre-determined formulas that may have limitations. It also helps to deepen our understanding of the concept of derivatives and their applications in mathematics and science.
While derivation by first principles is a powerful and fundamental method for finding derivatives, it can be time-consuming and tedious for more complex functions. In these cases, it may be more efficient to use other methods or pre-existing formulas to find the derivative.