Derivative of h(x) = sin(x)/x using the quotient rule

Let $h(x) = \dfrac{\sin(x)}{x}$ . Use the quotient rule, along with the derivative of $\sin(x)$ , to find $h'(x)$ .

Hint

Recall the quotient rule: if $h(x) = \dfrac{f(x)}{g(x)}$ , then $h'(x) = \dfrac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$ . Here $f(x) = \sin(x)$ and $g(x) = x$ , and you'll need $\dfrac{d}{dx}[\sin x] = \cos x$ .

Step-by-step solution

Identify $f(x) = \sin(x)$ and $g(x) = x$ , so $f'(x) = \cos(x)$ and $g'(x) = 1$ . The quotient rule states $h'(x) = \dfrac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$ .

Substituting gives $h'(x) = \dfrac{x\cos(x) - \sin(x)\cdot 1}{x^2}$ . This simplifies to $h'(x) = \dfrac{x\cos(x) - \sin(x)}{x^2}$ .

Derivative of h(x) = sin(x)/x using the quotient rule

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