Textbook Question
Find an equation of the line tangent to the given curve at a.
y = 2x2 / (3x - 1); a = 1
3. Techniques of Differentiation
Product and Quotient Rules
6:23 minutesProblem 3.4.49
Textbook Question
Derivatives Find and simplify the derivative of the following functions.
g(w) = √w+w / √w-w
Verified step by step guidance1
Step 1: Rewrite the function \( g(w) = \frac{\sqrt{w} + w}{\sqrt{w} - w} \) in a form that is easier to differentiate. Consider using the quotient rule, which states that if you have a function \( \frac{u}{v} \), its derivative is \( \frac{u'v - uv'}{v^2} \).
Step 2: Identify \( u = \sqrt{w} + w \) and \( v = \sqrt{w} - w \). Find the derivatives \( u' \) and \( v' \). For \( u = \sqrt{w} + w \), use the derivative rules: \( u' = \frac{1}{2\sqrt{w}} + 1 \). For \( v = \sqrt{w} - w \), use: \( v' = \frac{1}{2\sqrt{w}} - 1 \).
Step 3: Apply the quotient rule: \( g'(w) = \frac{(\frac{1}{2\sqrt{w}} + 1)(\sqrt{w} - w) - (\sqrt{w} + w)(\frac{1}{2\sqrt{w}} - 1)}{(\sqrt{w} - w)^2} \).
Step 4: Simplify the expression in the numerator by distributing and combining like terms. Carefully expand each term and simplify.
Step 5: Simplify the entire expression by combining like terms and reducing the fraction if possible. Ensure that the final expression is in its simplest form.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivatives
A derivative represents the rate at which a function changes at a given point. It is defined as the limit of the average rate of change of the function as the interval approaches zero. In calculus, derivatives are used to find slopes of tangent lines, optimize functions, and analyze motion. The notation for the derivative of a function f is often written as f'(x) or df/dx.
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Derivatives
Quotient Rule
The Quotient Rule is a method for finding the derivative of a function that is the ratio of two other functions. If you have a function h(x) = f(x)/g(x), the derivative h'(x) is given by (g(x)f'(x) - f(x)g'(x)) / (g(x))^2. This rule is essential when differentiating functions that involve division, ensuring that both the numerator and denominator are correctly accounted for in the derivative.
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Simplification of Derivatives
Simplifying derivatives involves reducing the expression to its simplest form after differentiation. This can include factoring, canceling common terms, or combining like terms. Simplification is crucial for making the derivative easier to interpret and use in further calculations, such as finding critical points or analyzing the behavior of the function.
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