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We use the chain rule to unleash the derivatives of the trigonometric functions.
Up until this point of the course we have been ignoring a large class of functions:
Trigonometric functions other than . We know that Armed with this fact we will
discover the derivatives of all of the standard trigonometric functions.
The derivative of cosine
Now that we know the derivative of cosine, we may
combine this with the chain rule, so we have that and so
Next we have:
The derivative of tangent
We’ll rewrite as and use the quotient rule. Write with me:
Applying the quotient rule, and the product rule, and the derivative of
Finally, we have:
The derivative of secant
We’ll rewrite as and use the power rule and the chain rule. Write
The derivatives of the cotangent and cosecant are similar and left as exercises.
Putting this all together, we have:
The Derivatives of Trigonometric Functions
Applying the product rule the facts above, we know that and so
When working with derivatives of trigonometric functions, we suggest you use
radians for angle measure. For example, while one must be careful with derivatives
as Alternatively, one could think of as meaning , as then . In this case