Question

Find the derivatives of the functions in Exercises 19–40.y = (5 − 2x)⁻³ + (1 / 8)(2 / x + 1)⁴

Accepted Answer

Step 1: Identify the structure of the function. The given function is y = (5 − $$2x)^{−3} + (1/8)(2/x$$ + $$1)^{4}. It $$consists of two terms: the first term is a power function, and the second term is a composite function involving a constant multiplier and a power of a rational expression. Step 2: Differentiate the first term, (5 − $$2x)^{−3}$$, using the chain rule. Let u = (5 − 2x). Then, the derivative of $$u^{−3} is$$ −$$3u^{−4}$$ multiplied by the derivative of u, which is −2. Combine these results to find the derivative of the first term. Step 3: Differentiate the second term, (1/8)(2/x + $$1)^{4}$$, using the chain rule. First, treat (1/8) as a constant multiplier. Let v = (2/x + 1). The derivative of $$v^{4} is$$ $$4v^{3}$$ multiplied by the derivative of v. To find the derivative of v, rewrite 2/x as $$2x^{−1}$$ and differentiate it. Step 4: Combine the results from Steps 2 and 3. Add the derivatives of the two terms together to form the derivative of the entire function y. Step 5: Simplify the expression for the derivative by combining like terms and ensuring the result is in its simplest form. This may involve rewriting negative exponents or rational expressions for clarity.

Find the derivatives of the functions in Exercises 19–40.

y = (5 − 2x)⁻³ + (1 / 8)(2 / x + 1)⁴

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Key Concepts

Derivative

Chain Rule

Power Rule