Question

Use a computer algebra system to differentiate the function. $$ f(\theta)=\frac{\sin \theta}{1-\cos \theta} $$

Short answer

The derivative of the given function is \[f'(\theta) = \frac{1}{(1-\cos \theta)^2}\]

Step-by-step solution

Find the Derivative of the Numerator and Denominator

First identify the numerator and denominator of the fraction. Here, \(\sin \theta\) is the numerator and \(1-\cos \theta\) is the denominator. The derivative of the numerator \(\sin \theta\) is \(\cos \theta\) and the derivative of the denominator \(1-\cos \theta\) is \(-\sin \theta\).

Apply the Quotient Rule

Then, apply the quotient rule to this fraction. Therefore, the derivative \(f'(\theta)\) is given by: \[f'(\theta) = \frac{(\cos \theta)(1-\cos \theta)-(-\sin \theta)(\sin \theta)}{(1-\cos \theta)^2}\]

Simplify the Derivative

Lastly, simplify the derivative to its simplest form. By simplifying the expression in the numerator, we find: \[f'(\theta) = \frac{1-\cos^2 \theta+\sin^2 \theta}{(1-\cos \theta)^2}\]. By using the Pythagorean trigonometric identity \(\sin^2 \theta+\cos^2 \theta = 1\), the numerator becomes \(1\). Thus the final derivative is: \[f'(\theta) = \frac{1}{(1-\cos \theta)^2}\]