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Apuntes

• Subtopic: Derivatives of Trigonometric and Exponential Functions 22. $f(x)=\frac{1+\sin^2x}{1+\cos^2x}$ ⇒ $f'(x)=\frac{6 \sin x \cos x}{(1+\cos^2x)^2}$ 23. $f(x)=x^2 \cdot e^{x^3}$ ⇒ $f'(x)=x \cdot e^{x^3} \, (2+3x^3)$ 24. $f(x)=e^{\sqrt{x^2+1}}$ ⇒ $f'(x)=x \cdot e^{\sqrt{x^2+1}} \cdot \frac{1}{\sqrt{x^2+1}}$ 25. $f(x)=x \cdot e^{-1/x^2}$ ⇒ $f'(x)=e^{-1/x^2} \left(1+\frac{2}{x^2}\right)$ 26. $f(x)=\frac{1-\cos(x^2)}{1+\cos(x^2)}$ ⇒ $f'(x)=\frac{4x \sin x^2}{(1+\cos(x^2))^2}$ 27. $f(x)=\ln \frac{x}{\sqrt{x^2+9}}$ ⇒ $f'(x)=\frac{-9}{x \, (x^2+9)}$ 28. $f(x)=\ln \frac{1-\cos x}{1+\cos x}$ ⇒ $f'(x)=\csc x$ • Subtopic: Derivatives of Inverse Trigonometric Functions 29. $f(x)=\arctan(x^2-1)$ ⇒ $f'(x)=\frac{2x}{x^4-2x^2+2}$ 30. $f(x)=\arctan \frac{1-x^2}{1+x^2}$ ⇒ $f'(x)=\frac{-2x}{1+x^4}$ 31. $f(x)=\arcsin(1+x) + \sqrt{2x-x^2}$ ⇒ $f'(x)=\frac{-x}{\sqrt{2x-x^2}}$ 32. $f(x)=\arccos x \frac{x}{\sqrt{1-x^2}}$ ⇒ $f'(x)=\frac{x \arccos x - \sqrt{1-x^2}}{x^2 \sqrt{1-x^2}}$ • Subtopic: Derivatives of Logarithmic and Power Functions 33. $f(x)=\ln \frac{\cos x}{x-1}$ ⇒ $f'(x)=\frac{1}{x^2} \cdot \tan \frac{x-1}{x}$ 34. $f(x)=\frac{1+\sqrt{\sin x}}{1-\sqrt{\...