Final answer:
The question addresses calculus techniques for finding the derivative of functions, including the limit definition of a derivative, the power rule, the product rule, and the quotient rule. These methods are vital in mathematics, especially in fields requiring dimensional analysis.
Step-by-step explanation:
The question pertains to different techniques in calculus for finding the derivative of functions. To find the derivative using the limit definition, one must evaluate the limit of the difference quotient. This process involves taking the limit as Δx approaches zero of the expression (f(x+Δx) - f(x))/Δx. Alternatively, the power rule for differentiation, which states that if f(x) = x^n, then f'(x) = nx^(n-1), can be applied directly for functions that are powers of x. When dealing with functions that are products of two or more functions, the product rule must be employed, which states that (fg)' = f'g + fg'. Lastly, for functions that are quotients of two other functions, the quotient rule is used, which provides that (f/g)' = (f'g - fg') / g^2.
Note that for polynomials with multiple term sums, the power rule can be applied to each term individually and the derivatives summed. When considering physical quantities and dimensional analysis in calculus, it's important to understand that the dimensions of a derivative will be the ratio of the dimensions of the numerator to those of the denominator. This is important in physics and engineering where dimensions have physical meaning.