FAC1004 Tutorial 7 — Derivatives of Inverse Trigonometric & Hyperbolic Functions
Centre for Foundation Studies in Science
University of Malaya
FAC1004 Advanced Mathematics 2
Topic: Derivative of Inverse Trigonometric Function
Question 1
Consider the following inverse trigonometric equation:
$$\sin^{-1} 2x + \frac{\pi}{4} = \tan^{-1}\frac{x}{\sqrt{1-x^2}}$$
Solve and show that the solution that satisfies the equation is:
$$x = \frac{-1}{2\sqrt{10} - 4\sqrt{2}}$$
Question 2
Show that:
(a) $\frac{d}{dx}\left[\cos^{-1} x\right] = -\frac{1}{\sqrt{1-x^2}}$
(b) $\frac{d}{dx}\left[\tan^{-1} x\right] = \frac{1}{1+x^2}$
(c) $\frac{d}{dx}\left[\csc^{-1} x\right] = -\frac{1}{|x|\sqrt{x^2-1}}$
Question 3
Find the first derivative of the following functions:
(a) $y = \sin^{-1}(3x)$
(b) $y = \cos^{-1}\left(\frac{x+1}{2}\right)$
(c) $y = \sin^{-1}\left(\frac{1}{x}\right)$
(d) $y = \sec^{-1}(x^4)$
(e) $y = \tan^{-1}(e^{3x})$
Question 4
Differentiate the following functions:
(a) $y = x + \sin^{-1}(e^{-3x})$
(b) $y = \ln(x^2) \sec^{-1}(4x^2)$
(c) $y = \tan^{-1}(x^2) \csc^{-1}(\ln x)$
(d) $y = \frac{\cos^{-1} 2x}{3x - e^{2x}}$
(e) $y = \frac{e^{3x} \sin^{-1}(5x)}{\ln(x^2) \tan x}$
Question 5
Show that $\cosh x + \sinh x = e^x$.
Question 6
Prove that $\tanh x = \frac{e^x - e^{-x}}{e^x + e^{-x}}$ and $\text{sech} x = \frac{2}{e^x + e^{-x}}$.
Hence, verify the identity $\text{sech}^2 x = 1 - \tanh^2 x$.
Key Concepts Covered
- Derivatives of Inverse Trigonometric Functions — Standard formulas
- Chain Rule — Differentiating composite functions
- Product Rule — Differentiating products of functions
- Quotient Rule — Differentiating quotients of functions
- Hyperbolic Functions — Definitions and identities
- Exponential Form — Relationship between hyperbolic and exponential functions