Mathemateical Methods of Physics

Supplimentary

Derivative

Trigonometry

• \(\frac{d}{dx} \sin(ax) = a \cos(ax)\)
• \(\frac{d}{dx} \cos(ax) = -a \sin(ax)\)
• \(\frac{d}{dx} \tan(ax) = a \sec^2(ax)\)
• \(\frac{d}{dx} \csc(ax) = -a \csc(ax) \cot(ax)\)
• \(\frac{d}{dx} \cot(ax) = -a \csc^2(ax)\)
• \(\frac{d}{dx} \sec(ax) = a \sec(ax) \tan(ax)\)

Inverse Trigonometry

• \(\frac{d}{dx} \sin^{-1}(ax) = \frac{a}{\sqrt{1 - (ax)^2}}\)
• \(\frac{d}{dx} \cos^{-1}(ax) = -\frac{a}{\sqrt{1 - (ax)^2}}\)
• \(\frac{d}{dx} \tan^{-1}(ax) = \frac{a}{1 + (ax)^2}\)
• \(\frac{d}{dx} {\cot^{-1}}(ax) = -\frac{a}{1 + (ax)^2}\)
• \(\frac{d}{dx} {\sec^{-1}}(ax) = \frac{a}{|ax| \sqrt{(ax)^2 - 1}}\)
• \(\frac{d}{dx} {\csc^{-1}}(ax) = -\frac{a}{|ax| \sqrt{(ax)^2 - 1}}\)

Hyperbolic

• \(\frac{d}{dx}\sinh(ax) = a\cosh(ax)\)
• \(\frac{d}{dx}\cosh(ax) = a\sinh(ax)\)
• \(\frac{d}{dx}\tanh(ax) = a \ \mbox{sech}^2(ax)\)

Inverse Hyperbolic

• \(\frac{d}{dx} \sinh^{-1}(ax) = \frac{a}{\sqrt{1 + (ax)^2}}\)
• \(\frac{d}{dx} \cosh^{-1}(ax) = \frac{a}{\sqrt{(ax)^2 - 1}}\)
• \(\frac{d}{dx} \tanh^{-1}(ax) = \frac{1}{1 - (ax)^2}\)