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Chapter 5: Problem 45

Find the derivative of the function. \(g(x)=\frac{\arcsin 3 x}{x}\)

Short Answer

Expert verified
The derivative of the given function is \(g'(x) = \frac{3}{x\sqrt{1-9x^2}} - \frac{\arcsin 3x}{x^2}\)

Step by step solution

01

Identify the functions in the quotient

In the given function \(g(x)=\frac{\arcsin 3x}{x}\), we have two functions: \(f(x)=\arcsin 3x\) (in the numerator) and \(h(x)=x\) (in the denominator). The differentiation requires applying the quotient rule along with the chain rule.
02

Apply the Quotient Rule and Chain Rule

The Quotient Rule states that the derivative of \(\frac{f(x)}{h(x)}\) is \(\frac{h(x)\cdot f'(x) - f(x)\cdot h'(x)}{h^2(x)}\). The Chain rule is applied when differentiating function compositions, as is the case with \(f(x) = \arcsin 3x\). Here the inner function \(u(x)=3x\) and the outer function \(v(u)=\arcsin u\). Thus \(f'(x) = v'(u)\cdot u'(x)\). We know that the derivative of \(\arcsin u\) is \(\frac{1}{\sqrt{1-u^2}}\) and derivative of \(3x\) is \(3\). So, \(f'(x) = \frac{3}{\sqrt{1-(3x)^2}}\). The derivative of \(h(x) = x\) is simply \(h'(x) = 1\).
03

Substitute derivations into the Quotient Rule

Substituting these derivations into the quotient rule, we get \(g'(x) = \frac{x \cdot \frac{3}{\sqrt{1-(3x)^2}} - \arcsin 3x \cdot 1}{x^2}\)
04

Simplify the final expression

Simplifying this expression further, we get \(g'(x) = \frac{3}{x\sqrt{1-9x^2}} - \frac{\arcsin 3x}{x^2}\)

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