Use differentiation to verify each equation.
d/dx(x / √1−x²) = 1 / (1−x²)^3/2.

Verified step by step guidance

1. Step 1: Begin by identifying the function you need to differentiate, which is f(x) = x / √(1−x²). This is a quotient, so you'll use the quotient rule for differentiation.
2. Step 2: Recall the quotient rule: if you have a function f(x) = u(x)/v(x), then its derivative f'(x) = (u'(x)v(x) - u(x)v'(x)) / (v(x))². Here, u(x) = x and v(x) = √(1−x²).
3. Step 3: Differentiate u(x) = x to get u'(x) = 1. Next, differentiate v(x) = √(1−x²). Use the chain rule: v'(x) = (1/2)(1−x²)^(-1/2) * (-2x) = -x / √(1−x²).
4. Step 4: Substitute u(x), u'(x), v(x), and v'(x) into the quotient rule formula: f'(x) = (1 * √(1−x²) - x * (-x / √(1−x²))) / (√(1−x²))².
5. Step 5: Simplify the expression obtained in Step 4. The numerator becomes √(1−x²) + x²/√(1−x²), and the denominator is 1−x². Simplify further to verify that f'(x) = 1 / (1−x²)^(3/2).

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. The derivative represents the rate of change of the function with respect to its variable. It is essential for analyzing the behavior of functions, including their slopes and rates of growth or decay. Understanding the rules of differentiation, such as the product rule and chain rule, is crucial for solving problems involving derivatives.

Chain Rule

The chain rule is a specific technique in differentiation used to differentiate composite functions. It states that if a function is composed of two functions, say f(g(x)), the derivative can be found by multiplying the derivative of the outer function by the derivative of the inner function. This rule is particularly useful when dealing with functions that involve square roots or other nested operations, as seen in the given equation.

Simplifying Derivatives

Simplifying derivatives involves manipulating the expression obtained after differentiation to make it easier to interpret or compare with other expressions. This can include factoring, combining like terms, or rewriting the expression in a different form. In the context of the given equation, simplifying the derivative will help verify the correctness of the equation by ensuring both sides match after differentiation.