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3.4 Differentiating Inverse Trigonometric Functions

Syllabus
2020
Topic
3.4
Level
—

Learning objectives

—FUN-3.E—Calculate derivatives of inverse and inverse trigonometric functions—Topic 3.4• FUN-3.E Calculate derivatives of inverse and inverse trigonometric functions.• FUN-3.E.2 The chain rule applied with the definition of an inverse function, or the formula for the derivative of an inverse function, can be used to find the derivatives of inverse trigonometric functions.• Enduring understanding FUN-3: Recognizing opportunities to apply derivative rules can simplify differentiation.

Differentiate Inverse Trig Functions

Inverse trigonometric functions return angles. Their derivative formulas reflect the reciprocal-rate rule for inverse functions. If the input is a differentiable expression u(x)u(x), every formula also includes the inner derivative u′(x)u'(x).

\frac{d}{dx}(\arcsin u)=\frac{u'}{\sqrt{1-u^2}},\qquad \frac{d}{dx}(\arccos u)=-\frac{u'}{\sqrt{1-u^2}},\qquad \frac{d}{dx}(\arctan u)=\frac{u'}{1+u^2}

\frac{d}{dx}(\operatorname{arccot}u)=-\frac{u'}{1+u^2},\qquad \frac{d}{dx}(\operatorname{arcsec}u)=\frac{u'}{|u|\sqrt{u^2-1}},\qquad \frac{d}{dx}(\operatorname{arccsc}u)=-\frac{u'}{|u|\sqrt{u^2-1}}

  1. Identify the inverse trigonometric function and its input u(x)u(x).\n2. Write the matching derivative formula.\n3. Substitute u(x)u(x) everywhere in the formula.\n4. Multiply the numerator by u′(x)u'(x) and simplify without losing the required sign.

For y=arctan⁡(2x−1)y=\arctan(2x-1), let u=2x−1u=2x-1, so u′=2u'=2. Then dydx=u′1+u2=21+(2x−1)2.\frac{dy}{dx}=\frac{u'}{1+u^2}=\frac{2}{1+(2x-1)^2}. The factor 22 records how quickly the inner expression changes.

sin⁡−1x\sin^{-1}x means arcsin⁡x\arcsin x, not 1/sin⁡x1/\sin x. Remember the negative signs for arccos⁡\arccos and arccot⁡\operatorname{arccot}, and the absolute value in the arcsec⁡\operatorname{arcsec} and arccsc⁡\operatorname{arccsc} denominators. For finite real derivatives, ∣u∣<1|u|<1 in the arcsine/arccosine formulas and ∣u∣>1|u|>1 in the arcsec/arccsc formulas.