Derivatives Of Trigonometric Functions

Source: http://wmueller.com/precalculus/newfunc/images/triggraphs.gif

This post will be about derivatives of trigonometric functions. It is assumed that the reader knows about \sin(x), \cos(x), and \tan(x) along with the reciprocal trigonometric functions \sec(x), \cot(x) and \csc(x). Knowledge of the chain rule is not assumed here.


Table Of Contents

  1. Derivatives of Trigonometric Functions
  2. Memory Aids
  3. Examples
  4. Notes


Derivatives of Trigonometric Functions

When it comes to derivatives of trigonometric functions, it is mostly memorizing. Here is the list of the derivatives.

1)    \displaystyle \dfrac{d}{dx} \sin(x) = \cos(x)

2)    \displaystyle \dfrac{d}{dx} \cos(x) = - \sin(x)

3)    \displaystyle \dfrac{d}{dx} \tan(x) = \sec^{2}(x)

4)    \displaystyle \dfrac{d}{dx} \sec(x) = \sec(x) \cdot \tan(x)

5)    \displaystyle \dfrac{d}{dx} \csc(x) = - \csc(x) \cdot \cot(x)

6)    \displaystyle \dfrac{d}{dx} \cot(x) = - \csc^{2}(x)

The proofs of the derivatives are not provided here. They are somewhat technical. One can find such proofs at http://www.math.brown.edu/UTRA/trigderivs.html.


Memory Aids 

If memorizing the six derivatives is a bit of a hassle, consider these memory aids.

  1. The ones that start with the letter c have a negative in their derivatives. (I found this very useful.)
  2. Derivatives of \tan(x) and \cot(x)} have squares.
  3. Note that \sin(x) and \cos}(x) are alternating with each other. Watch for that pesky negative sign!
  4. The derivatives of \sec(x) and \csc(x) have two terms in the product with one of them being itself (respectively).
  5. You could use abbreviations such as sc, cns, ts2, sst, cncc and cc2 or something.


Examples

Here are some examples.

Example One

\displaystyle f(x) = -\cos(x)

\displaystyle \begin{array} {lcl} f'(x) & = & - \dfrac{d}{dx} \cos(x) \\ & = & - ( - \sin(x) )\\ & = & \sin(x) \\ \end{array} \\

Example Two (Product Rule)

\displaystyle f(x) = \sin(x) \cdot \tan(x)

\displaystyle \begin{array} {lcl} f'(x) & = & \dfrac{d}{dx} \sin(x) \cdot \tan(x) + \sin(x) \cdot \dfrac{d}{dx} \tan(x) \\ & = & \cos(x) \cdot \tan(x) + \sin(x) \cdot \sec^{2}(x) \\ & = & \cos(x) \cdot \dfrac{\sin(x)}{\cos(x)} + \sin(x) \cdot \sec^{2}(x) \\ & = & \sin(x) + \sin(x) \cdot \sec^{2}(x) \\ & = & \sin(x) \cdot (1 + \sec^{2}(x)) \\ \end{array}

Example Three (Quotient Rule)

\displaystyle f(x) = \dfrac{\sin(x)}{\cos(x)}

\displaystyle \begin{array} {lcl} f'(x) & = & \dfrac{\dfrac{d}{dx} \sin(x) \cdot \cos(x) - \sin(x) \cdot \dfrac{d}{dx} \cos(x)}{\cos^{2}(x)}\\ & = & \dfrac{\cos^{2}(x) - \sin(x) \cdot (- \sin(x)) }{\cos^{2}(x)}\\ & = & \dfrac{\cos^{2}(x) + \sin^{2}(x)}{\cos^{2}(x)}\\ & = & 1 + \dfrac{\sin^{2}(x)}{\cos^{2}(x)}\\ & = & 1 + (\dfrac{\sin(x)}{\cos(x)})^{2}\\ & = & 1 + \tan^{2}(x) \\ \end{array}


Notes

Dealing with trigonometric functions and derivatives can be tricky. When you add in chain rule, product rule and quotient rule, one has to be really careful.

The featured image is taken from http://wmueller.com/precalculus/newfunc/images/triggraphs.gif.

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