Derivatives in Differential Form

In Exercises 17–28, find dy.

y = x√(1 − x²)

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Step 1: Identify the function y = x√(1 − x²). This is a product of two functions: x and √(1 − x²).

Step 2: Apply the product rule for differentiation, which states that if you have a function y = u*v, then dy/dx = u'(v) + u(v'). Here, let u = x and v = √(1 − x²).

Step 3: Differentiate u = x with respect to x, which gives u' = 1.

Step 4: Differentiate v = √(1 − x²) with respect to x. Use the chain rule: v' = (1/2)(1 − x²)^(-1/2) * (-2x). Simplify this expression.

Step 5: Substitute u', v, u, and v' into the product rule formula: dy/dx = u'(v) + u(v'). Simplify the expression to find dy/dx.

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Derivative

The derivative of a function measures how the function's output changes as its input changes. It is a fundamental concept in calculus, representing the rate of change or slope of the function at a given point. For a function y = f(x), the derivative is denoted as dy/dx and can be found using various rules such as the power rule, product rule, and chain rule.

Product Rule

The product rule is used to find the derivative of a product of two functions. If y = u(x)v(x), where both u and v are functions of x, the derivative dy/dx is given by u'(x)v(x) + u(x)v'(x). This rule is essential when dealing with functions that are multiplied together, as in the given problem y = x√(1 − x²).

Chain Rule

The chain rule is a method for finding the derivative of a composite function. If a function y = f(g(x)) is composed of two functions, the derivative dy/dx is found by multiplying the derivative of the outer function by the derivative of the inner function: f'(g(x))g'(x). This rule is crucial when differentiating functions like √(1 − x²), where the inner function is 1 − x².

Derivatives in Differential FormIn Exercises 17–28, find dy.y = x√(1 − x²)