Derivatives Of Trig Functions

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Derivatives of Tangent, Cotangent, Secant, and Cosecant

We can get the derivatives of the other four trig functions by applying the quotient rule to sine and cosine. For instance, \begin{eqnarray*} \frac{d }{dx}\big( \tan(x)\big) & = & \left (\frac{\sin(x)}{\cos(x)} \right )' \cr & = & \frac{\cos(x) (\sin(x))' - \sin(x) (\cos(x))'}{\cos^2(x)} \cr & = & \frac{\cos^2(x) + \sin^2(x)}{\cos^2(x)} \cr & = & \frac{1}{\cos^2(x)} \cr &=& \sec^2(x). \end{eqnarray*}

Before watching the video, try one yourself:

DO: Using the reciprocal trig relationships to turn the secant into a function of sine and/or cosine, and also use the derivatives of sine and/or cosine, to find $\displaystyle\frac{d}{dx}\sec x$.

You must know all of the following derivatives. Notice that you really need only learn the left four, since the derivatives of the cosecant and cotangent functions are the negative "co-" versions of the derivatives of secant and tangent. Notice also that the derivatives of all trig functions beginning with "c" have negatives.
$$\begin{array}{ll} \frac{d}{dx}\sin(x)=\cos (x)\\ \frac{d}{dx}\cos(x)=-\sin(x)\\ \frac{d}{dx}\tan(x) = \sec^2(x) &\frac{d}{dx}\cot(x)=-\csc^2(x)\\ \frac{d}{dx}\sec(x) =\sec(x) \tan(x) &\frac{d}{dx}\csc(x)=-\csc(x)\cot(x) \end{array}$$
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