Section 3.8
Derivatives of Inverse Trig Functions
Surprise! The derivatives of inverse trigonometric functions don't involve trig at all—they are purely algebraic.
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The Formulas
These arise from implicit differentiation and the Pythagorean identity .
Primary Functions
Inverse Sine
Inverse Tangent
Inverse Secant
"Co-" Functions (Negative)
Inverse Cosine
Inverse Cotangent
Inverse Cosecant
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Worked Example
Differentiate .
Note: We use the Product Rule because we are multiplying two functions of .
Step 1: Identify Parts
Step 2: Apply Product Rule ()
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More Examples
Quotient Rule
Differentiate .
Identify components:
Apply Quotient Rule:
Linear Combination
Differentiate .
Identify rules:
Just differentiate each term separately and keep the constants!
Assemble:
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Interactive Visualization
Inverse Tangent & The Witch of Agnesi
The derivative of arctan(x) sketches out a famous curve called the Witch of Agnesi (roughly). Notice it approaches 0 as x goes to infinity.
Original Function f(x)
Tangent Line
Derivative Function f'(x)
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Practice Quiz
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