Derivatives of products and quotients Find the derivative of the following functions by first expanding or simplifying the expression. Simplify your answers.
h(x) = √x (√x-x³/²)

검증된 단계별 안내

Step 1: Begin by expanding the expression h(x) = \(\sqrt{x}\)(\(\sqrt{x}\) - x^{3/2}). Distribute \(\sqrt{x}\) across the terms inside the parentheses.

Step 2: Simplify the expression by multiplying \(\sqrt{x}\) with each term inside the parentheses: \(\sqrt{x}\) \(\cdot\) \(\sqrt{x}\) - \(\sqrt{x}\) \(\cdot\) x^{3/2}.

Step 3: Recognize that \(\sqrt{x}\) \(\cdot\) \(\sqrt{x}\) = x^{1/2} \(\cdot\) x^{1/2} = x^{1/2 + 1/2} = x^1 = x.

Step 4: Simplify the second term: \(\sqrt{x}\) \(\cdot\) x^{3/2} = x^{1/2} \(\cdot\) x^{3/2} = x^{1/2 + 3/2} = x^2.

Step 5: Combine the simplified terms to get h(x) = x - x^2. Now, find the derivative h'(x) by differentiating each term separately: h'(x) = \(\frac{d}{dx}\)(x) - \(\frac{d}{dx}\)(x^2).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Product Rule

The Product Rule is a fundamental principle in calculus used to find the derivative of the product of two functions. If u(x) and v(x) are two differentiable functions, the derivative of their product is given by (u*v)' = u'v + uv'. This rule is essential when dealing with functions that are multiplied together, as it allows for the correct application of differentiation.

Quotient Rule

The Quotient Rule is another key differentiation technique used when finding the derivative of a function that is the ratio of two other functions. If u(x) and v(x) are differentiable functions, the derivative of their quotient is given by (u/v)' = (u'v - uv')/v². This rule is crucial for simplifying expressions where one function is divided by another.

Simplification of Expressions

Simplification of expressions involves rewriting a mathematical expression in a more manageable or understandable form. In calculus, this often includes factoring, expanding, or combining like terms to make differentiation easier. For the function h(x) = √x (√x - x³/²), simplifying the expression before applying differentiation rules can lead to a more straightforward calculation of the derivative.

Derivatives of products and quotients Find the derivative of the following functions by first expanding or simplifying the expression. Simplify your answers. h(x) = √x (√x-x³/²)