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Chapter 7: Problem 65

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ y=\left(\frac{6-5 x}{x^{2}-1}\right)^{2} $$

Short Answer

Expert verified
The derivative of the function is \(y'=2(\frac{6-5x}{x^2-1}) \cdot (\frac{-5(x^2-1)-(6-5x)(2x)}{(x^2-1)^2})\)

Step by step solution

01

Apply Chain Rule

To handle the square of the whole fraction, use the chain rule. The chain rule states that the derivative of \((f(g(x)))') = f'(g(x)) \cdot g'(x)\). Here, consider \(f(u) = u^2\) and \(u= g(x) = \frac{6-5x}{x^2-1}\). So, the derivative of \(y\) at this step becomes: \(y' = 2u \cdot u'\), where \(u'\) is the derivative of \(u\) with respect to \(x\), which needs to be computed using the quotient rule.
02

Apply Quotient Rule

To find \(u'\), apply the quotient rule to \(\frac{6-5x}{x^2-1}\). The quotient rule is defined as: \((\frac{f}{g})' = \frac{f'g - fg'}{g^2}\). Here, \(f = 6-5x\) and \(g = x^2 - 1\). From this, we can find that \(f' = -5\) and \(g' = 2x\). Thus, we can compute \(u'\) as : \(u' = \frac{f'g - fg'}{g^2} = \frac{-5(x^2 - 1) - (6-5x)(2x)}{(x^2 - 1)^2}\)
03

Substitute \(u'\) Back

Now we need to substitute \(u'\) back into the derivative found in step 1, to get the final derivative of our original function \(y\). So the final derivative \(y'\) becomes: \(y'=2(\frac{6-5x}{x^2-1}) \cdot (\frac{-5(x^2-1)-(6-5x)(2x)}{(x^2-1)^2})\)

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Most popular questions from this chapter

Use a symbolic differentiation utility to find the derivative of the function. Graph the function and its derivative in the same viewing window. Describe the behavior of the function when the derivative is zero. $$ f(x)=\sqrt{x}\left(2-x^{2}\right) $$

Find the point(s), if any, at which the graph of \(f\) has a horizontal tangent. $$ f(x)=\frac{x^{2}}{x^{2}+1} $$

Use the General Power Rule to find the derivative of the function. $$ g(x)=(4-2 x)^{3} $$

The ordering and transportation cost \(C\) per unit for the components used in manufacturing a product is \(C=\left(375,000+6 x^{2}\right) / x, \quad x \geq 1\) where \(C\) is measured in dollars and \(x\) is the order size. Find the rate of change of \(C\) with respect to \(x\) when (a) \(x=200\), (b) \(x=250\), and (c) \(x=300\). Interpret the meaning of these values.

Identify the inside function, \(u=g(x)\), and the outside function, \(y=f(u)\). $$ y=f(g(x)) \quad u=g(x) \quad y=f(u) $$ $$ y=(3 x+1)^{-1} $$
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