Problem 45 - Calculus of a Single Variable

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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 \(h'(x) = 1\).

Step 3: Substitute derivations into the Quotient Rule

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

Step 4: Simplify the final expression

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