Calculate the derivative of the following functions.
y = √x+√x+√x

Verified step by step guidance

1. Step 1: Recognize that the function y = \sqrt{x} + \sqrt{x} + \sqrt{x} can be simplified to y = 3\sqrt{x}.
2. Step 2: Recall the power rule for derivatives, which states that if y = x^n, then the derivative y' = nx^{n-1}.
3. Step 3: Rewrite \sqrt{x} as x^{1/2} to apply the power rule. Therefore, y = 3x^{1/2}.
4. Step 4: Differentiate y = 3x^{1/2} using the power rule. The derivative of x^{1/2} is (1/2)x^{-1/2}.
5. Step 5: Multiply the derivative of x^{1/2} by the constant 3 to get the final derivative: y' = 3 * (1/2)x^{-1/2}.

Key Concepts

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In calculus, the derivative is often denoted as f'(x) or dy/dx, and it provides critical information about the function's behavior, such as its slope and concavity.

Chain Rule

The chain rule is a fundamental technique for finding the derivative of composite functions. It states that if a function y is composed of two functions u and x (i.e., y = f(u) and u = g(x)), then the derivative of y with respect to x can be found by multiplying the derivative of f with respect to u by the derivative of g with respect to x. This rule is essential when differentiating functions that involve nested expressions.

Power Rule

The power rule is a basic rule for differentiating functions of the form f(x) = x^n, where n is a real number. According to this rule, the derivative f'(x) is given by n*x^(n-1). This rule simplifies the process of differentiation for polynomial and root functions, making it easier to compute derivatives quickly and efficiently.