Derivatives of Inverse Functions

1. Derivative of Arccosine

ddx[cos⁡−1(u)]=−11−u2⋅u′\frac{d}{dx}[\cos^{-1}(u)] = -\frac{1}{\sqrt{1-u^2}} \cdot u'

Definition: The derivative of arccosine is negative because cosine decreases on its principal interval. It involves the square root of 1−u21 - u^2 in the denominator.

2. Derivative of Arcsine

ddx[sin⁡−1(u)]=11−u2⋅u′\frac{d}{dx}[\sin^{-1}(u)] = \frac{1}{\sqrt{1-u^2}} \cdot u'

Definition: The derivative of arcsine is positive and has the same denominator as arccosine. It applies when ∣u∣<1|u| < 1.

3. Derivative of Arctangent

ddx[tan⁡−1(u)]=11+u2⋅u′\frac{d}{dx}[\tan^{-1}(u)] = \frac{1}{1+u^2} \cdot u'

Definition: The derivative of arctangent has a rational denominator 1+u21+u^2, which is always positive.

4. Derivative of Arcsecant

ddx[sec⁡−1(u)]=1∣u∣u2−1⋅u′\frac{d}{dx}[\sec^{-1}(u)] = \frac{1}{|u|\sqrt{u^2-1}} \cdot u'

Definition: The derivative of arcsecant includes both ∣u∣|u| and u2−1\sqrt{u^2-1}. This ensures correctness across positive and negative values of uu.

5. Derivative of Arccotangent

ddx[cot⁡−1(u)]=−11+u2⋅u′\frac{d}{dx}[\cot^{-1}(u)] = -\frac{1}{1+u^2} \cdot u'

Definition: Similar to arctangent but negative, because cotangent decreases in its domain.

6. Derivative of Arccosecant

ddx[csc⁡−1(u)]=−1∣u∣u2−1⋅u′\frac{d}{dx}[\csc^{-1}(u)] = -\frac{1}{|u|\sqrt{u^2-1}} \cdot u'

Definition: Similar to arcsecant but negative. It also requires the absolute value of uu and u2−1\sqrt{u^2-1}.