Find the first and second derivatives of the functions in Exercises 33–38.

w = ((1 + 3z) / 3z) (3 − z)

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Step 1: Begin by simplifying the function w = ((1 + 3z) / 3z) (3 − z). Distribute the terms to express w as a single fraction or polynomial if possible.

Step 2: Once simplified, apply the product rule to find the first derivative. The product rule states that if you have a function w = u(z) * v(z), then the derivative w' = u'(z) * v(z) + u(z) * v'(z). Identify u(z) and v(z) from your expression.

Step 3: Calculate the derivatives u'(z) and v'(z) separately. For u(z) = (1 + 3z) / 3z, use the quotient rule: if u(z) = f(z)/g(z), then u'(z) = (f'(z)g(z) - f(z)g'(z)) / (g(z))^2. For v(z) = (3 - z), simply find the derivative with respect to z.

Step 4: Substitute u'(z), v(z), u(z), and v'(z) back into the product rule formula to find the first derivative w'. Simplify the expression if necessary.

Step 5: To find the second derivative, differentiate w' with respect to z again. Use the appropriate rules (product, quotient, or chain rule) as needed, and simplify the resulting expression to obtain the second derivative.

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주요 개념

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Product Rule

The product rule is a fundamental technique in calculus used to find the derivative of a product of two functions. If you have two functions, u(z) and v(z), the derivative of their product is given by (u'v + uv'). This rule is essential when differentiating expressions where two functions are multiplied together, as seen in the given function w = ((1 + 3z) / 3z) (3 − z).

Quotient Rule

The quotient rule is used to differentiate functions that are ratios of two differentiable functions. For a function expressed as a quotient u(z)/v(z), the derivative is (v(z)u'(z) - u(z)v'(z)) / (v(z))^2. This rule is crucial for finding the derivative of the first part of the given function, (1 + 3z) / 3z, which is a quotient.

Chain Rule

The chain rule is a method for differentiating composite functions. If a function y = f(g(z)) is composed of two functions, the derivative is f'(g(z)) * g'(z). This rule is important when dealing with nested functions, ensuring that each layer of the function is correctly differentiated, which may be necessary when simplifying the derivatives of the given function.

Find the first and second derivatives of the functions in Exercises 33–38.w = ((1 + 3z) / 3z) (3 − z)