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Chapter 7: Problem 61

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ y=x \sqrt{2 x+3} $$

Short Answer

Expert verified
The derivative of the function \(y=x \sqrt{2x+3}\) is \(y'(x) = \sqrt{2x+3} + \frac{x}{\sqrt{2x+3}}\). The Product Rule and the Chain Rule were used for finding the derivative.

Step by step solution

01

Identify the Functions

Given \(y = x \sqrt{2x+3}\), identify the two functions as \(f(x) = x\) and \(g(x) = \sqrt{2x+3}\)
02

Apply Product Rule

Apply the Product Rule which is \((f \cdot g)'=f'g+fg'\) for \(y=x \sqrt{2x+3}\). Now calculate the derivative of \(f(x) = x\), which is \(f'(x) = 1\).
03

Apply Chain Rule

The function \(g(x) = \sqrt{2x+3}\) is a composite function, so the Chain Rule will be used for differentiation. The Chain Rule is \((f(g(x)))' = f'(g(x)) \cdot g'(x)\). Rewrite the function in the form \(g(x) = (2x + 3)^{0.5}\), then apply the Chain Rule. Derive the outer function first keeping the inner function as is, then derive the inner function. Doing this gives us \(g'(x) = 0.5\cdot(2x+3)^{-0.5} \cdot 2\).
04

Combine the Results

Combine the derivatives of \(f(x)\) and \(g(x)\) according to the Product Rule. Thus, \(y' = f'g+fg' = 1 \cdot \sqrt{2x+3} + x \cdot 0.5 \cdot (2x+3)^{-0.5} \cdot 2\). Simplify this to get \(y'(x) = \sqrt{2x+3} + \frac{x}{\sqrt{2x+3}}\).

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Most popular questions from this chapter

Match the function with the rule that you would use to find the derivative most efficiently. (a) Simple Power Rule (b) Constant Rule (c) General Power Rule (d) Quotient Rule $$ f(x)=\frac{5}{x^{2}+1} $$

Find an equation of the tangent line to the graph of \(f\) at the point \((2, f(2)) .\) Use a graphing utility to check your result by graphing the original function and the tangent line in the same viewing window. $$ f(x)=x \sqrt{x^{2}+5} $$

Use the General Power Rule to find the derivative of the function. $$ y=\left(2 x^{3}+1\right)^{2} $$

The model \(f(t)=\frac{t^{2}-t+1}{t^{2}+1}\) measures the level of oxygen in a pond, where \(t\) is the time (in weeks) after organic waste is dumped into the pond. Find the rates of change of \(f\) with respect to \(t\) when (a) \(t=0.5,(\) b) \(t=2\), and (c) \(t=8\)

Use the given information to find \(f^{\prime}(2)\) \(g(2)=3\) and \(g^{\prime}(2)=-2\) \(h(2)=-1 \quad\) and \(\quad h^{\prime}(2)=4\) $$ f(x)=\frac{g(x)}{h(x)} $$
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