Derivatives of Trig Functions

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Trigonometry Review

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Inverse Trig Functions

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Rates of Change

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The Derivative Function

The derivative function
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Basic Differentiation Rules

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Product and Quotient Rules

The Product Rule
The Quotient Rule

Derivatives of Trig Functions

Necessary Limits
Derivatives of Sine and Cosine
Derivatives of Tangent, Cotangent, Secant, and Cosecant
Summary

The Chain Rule

Two Forms of the Chain Rule
Version 1
Version 2
Why does it work?
A hybrid chain rule

Implicit Differentiation

Introduction
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Derivatives of Inverse Trigs via Implicit Differentiation
A Summary

Derivatives of Logs

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Logarithmic Differentiation

Derivatives in Science

In Physics
In Economics
In Biology

Related Rates

Overview
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Example (ladder)
Example (shadow)

Linear Approximation and Differentials

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Examples
An example with negative $dx$

Differentiation Review

How to take derivatives
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Absolute and Local Extrema

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$f$ vs. $f'$

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The Second Derivative Test
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Newton's Method

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Anti-derivatives

Antiderivatives
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The Area under a curve

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Definite Integrals

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The Fundamental Theorem of Calculus

Three Different Concepts
The Fundamental Theorem of Calculus (Part 2)
The Fundamental Theorem of Calculus (Part 1)
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Indefinite Integrals and Anti-derivatives
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The NCT and Public Policy

Substitution

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Area Between Curves

Computation Using Integration
To Compute a Bulk Quantity
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Horizontal Slicing
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Volumes

Slicing and Dicing Solids
Solids of Revolution 1: Disks
Solids of Revolution 2: Washers
More Practice

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.

$$\begin{array}{ll}
\frac{d}{dx}\sin(x)=\cos (x)\\
\frac{d}{dx}\cos(x)=-\sin(x)\\
\frac{d}{dx}\tan(x) = \sec^2(x)
&\frac{d}{dx}\cot(x)=-\csc^2(x)\\
\frac{d}{dx}\sec(x) =\sec(x) \tan(x)
&\frac{d}{dx}\csc(x)=-\csc(x)\cot(x)
\end{array}$$