The derivative of the function f(x) = cos(√x) can be found using first principles, specifically the limit definition of the derivative: f'(x) = lim as h tends to zero of [f(x+h) - f(x)]/h. The challenge arises when evaluating cos(√(x+h)), which can be expressed as cos(√x(1 + h/x)^(1/2)). Utilizing the first two terms of the binomial expansion for (1 + h/x)^(1/2) simplifies the process. This method effectively addresses the complexities involved in differentiating composite functions.
PREREQUISITESStudents studying calculus, particularly those focusing on derivatives and the application of first principles in differentiation. This discussion is also beneficial for educators seeking to explain the concept of derivatives of composite functions.
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