Question

Extension of Example 8 Show that \(\frac{d}{d x} \cos \left(x^{\circ}\right)\) is \(-\frac{\pi}{180} \sin \left(x^{\circ}\right)\)

Short answer

Therefore, the derivative of \( \cos \left(x^{\circ}\right) \) is \( -\frac{\pi}{180} \sin \left(x^{\circ}\right) \).

Step-by-step solution

Transform Degrees to Radians

Firstly, recall that \(x^{\circ}\) represents x in degrees. To change to radians multiply by \(\frac{\pi}{180}\) since \(\pi\) radians is equal to 180 degrees. Thus we have \(x = \frac{\pi}{180} t\), where \(t\) is \(x^{\circ}\) in radians.

Apply Chain Rule

The Chain Rule in differentiation is applied as follows: \(\frac{d}{d x} \cos \left(t\right) = -\sin(t)\frac{d t}{d x}\). Apply this rule and remember to substitute for \(t\). Thus we have \(\frac{d}{d x} \cos \left(\frac{\pi}{180}t\right) = -\sin\left(\frac{\pi}{180}t\right)\frac{d}{d x}\left(\frac{\pi}{180}t\right)\). Also, remember \(\frac{d}{d x}\left(\frac{\pi}{180}t\right) = \frac{\pi}{180}\)

Final Calculation

Plugging in \(\frac{\pi}{180}\) for \(\frac{d}{d x}\left(\frac{\pi}{180}t\right)\) we obtain \(\frac{d}{d x} \cos \left(\frac{\pi}{180}t\right) = -\sin\left(\frac{\pi}{180}t\right)\times\frac{\pi}{180}\). Now, replace t with \(x^{\circ}\) to get \(\frac{d}{d x} \cos \left(x^{\circ}\right) = -\frac{\pi}{180}\sin \left(x^{\circ}\right)\).