Differentiating Trigonometric Functions

Find the derivative of each function:

1. \(f(x) =x^{2}\cos x\)
2. \(g(\theta ) =\dfrac{\cos \theta }{1-\sin \theta }\)
3. \(F(t) =\dfrac{e^{t}}{\cos t}\)

Solution

1. \[ \begin{eqnarray*} f^\prime (x) &=& \frac{d}{dx} (x^{2}\cos x) = x^{2}\frac{d}{dx}\cos x + (\frac{d}{dx} x^{2}) (\cos x)\\ &=& x^{2}(-\sin x) + 2x\cos x = 2x\cos x - x^{2}\sin x \end{eqnarray*} \]
2. \[ \begin{eqnarray*} g^\prime (\theta) &=& \frac{d}{d\theta} (\frac{\cos\theta}{1-\sin\theta}) = \frac{(\frac{d}{d\theta}\cos\theta) (1-\sin\theta) -(\cos\theta) [\frac{d}{d\theta}(1-\sin\theta) ]}{(1-\sin\theta)^{2}}\\ &=& \frac{-\sin\theta (1-\sin\theta) -\cos\theta (-\cos\theta)}{(1-\sin\theta)^{2}} =\frac{-\sin\theta +\sin^{2}\theta +\cos^{2}\theta}{(1-\sin\theta)^{2}}\\ &=&\frac{-\sin\theta+1}{(1-\sin\theta)^{2}}=\frac{1}{1-\sin\theta} \end{eqnarray*} \]
3. \[ \begin{eqnarray*} F\prime(t) &=&\frac{d}{dt}(\frac{e^{t}}{\cos t}) = \frac{(\frac{d}{dt}e^{t}) (\cos t) -e^{t}(\frac{d}{dt}\cos t)}{\cos^{2}t}=\frac{e^{t}\cos t-e^{t}(-\sin t) }{\cos ^{2}t}\\ &=&\frac{e^{t}(\cos t+\sin t)}{\cos^{2}t} \end{eqnarray*} \]