The derivative of a function f(x) at a point x is defined as the limit of the ratio of the change in the function value to the change in the input variable, as the change in the input variable approaches zero.
The formula for calculating the derivative of sin sq. root x is
f'(x) = (1/2)*cos(sqrt(x))/sqrt(x).
To calculate the derivative of sin sq. root x using the definition, you need to find the limit of the difference quotient as the change in x approaches 0. This involves plugging in h=0 and solving for the derivative using algebraic manipulations.
The steps for calculating the derivative of sin sq. root x are:
1. Write out the definition of a derivative.
2. Substitute sin sq. root x into the definition.
3. Simplify and manipulate the expression to get it in a form that can be evaluated at h=0.
4. Take the limit as h approaches 0 to find the derivative.
Yes, the derivative of sin sq. root x can be simplified further to f'(x) = cos(sqrt(x))/(2*sqrt(x)). This can be done by using trigonometric identities and algebraic manipulations.