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Chapter 3: Problem 2

How do you find the derivative of the quotient of two functions that are differentiable at a point?

Short Answer

Expert verified
Question: Using the Quotient Rule, find the derivative of the composite function f(x) = (2x^2 + 3x) / (x^2 - 4), and simplify your answer. Answer: To solve this problem, follow the steps outlined in the solution above: Step 1: Identify the functions u(x) and v(x). In this case, u(x) = 2x^2 + 3x, and v(x) = x^2 - 4. Step 2: Differentiate both u(x) and v(x). The derivative of u(x) is u'(x) = 4x + 3, and the derivative of v(x) is v'(x) = 2x. Step 3: Apply the Quotient Rule. Using the formula for the Quotient Rule, the derivative of f(x) is: f'(x) = (v(x) * u'(x) - u(x) * v'(x)) / (v(x))^2 = ((x^2 - 4) * (4x + 3) - (2x^2 + 3x) * (2x)) / (x^2 - 4)^2 Step 4: Simplify the expression. Expanding the terms and combining like terms, we get: f'(x) = (- 4x^3 - 7x^2 + 12) / (x^4 - 8x^2 + 16) Step 5: Since we are not asked to evaluate the derivative at a specific point, our final answer is: f'(x) = (- 4x^3 - 7x^2 + 12) / (x^4 - 8x^2 + 16)

Step by step solution

01

Identify the functions u(x) and v(x)

Assume that we have functions u(x) and v(x) that are differentiable at a point 'a'. The functions could be in any form, such as polynomials, trigonometric, or exponential, provided they are differentiable. Let us denote the composite function we want to find the derivative of as: f(x) = u(x) / v(x) where f(x) is the quotient of the functions u(x) and v(x).
02

Differentiate both u(x) and v(x)

Now we need to find the derivatives of u(x) and v(x) with respect to x. Let's denote their derivatives as u'(x) and v'(x). We can use any differentiation techniques required depending on the form of the functions u(x) and v(x).
03

Apply the Quotient Rule

Next, we apply the Quotient Rule to find the derivative of f(x). With the derivatives of u(x) and v(x) already computed, the Quotient Rule states that: f'(x) = (v(x) * u'(x) - u(x) * v'(x)) / (v(x))^2 Now simply plug in the expressions for u(x), v(x), u'(x), and v'(x) into the above formula to find the derivative of the quotient.
04

Simplify the expression (if necessary)

The resulting expression from applying the Quotient Rule may be a complex one. If possible, simplify the resulting expression by factoring, combining like terms, or using any other relevant algebraic techniques.
05

Evaluate derivative at a specific point (if required)

If the problem asks to find the derivative of the quotient at a specific point 'a', only then replace 'x' with 'a' in the simplified derivative expression and evaluate the result. Following this step-by-step process will allow you to find the derivative of the quotient of two functions that are differentiable at a point using the Quotient Rule.

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