Derivatives of Transcendental Functions

Derivatives of Transcendental Functions

Transcendental functions (like sine, cosine, $e^x$, $\ln x$) are fundamental to modeling periodic phenomena, growth, decay, and many engineering applications.

Inverse Trigonometric Functions

The inverse trigonometric functions ($\arcsin x$, $\arccos x$, etc.) have their own differentiation rules derived implicitly.

1. $\frac{d}{dx}[\arcsin u] = \frac{u'}{\sqrt{1-u^2}}$
2. $\frac{d}{dx}[\arccos u] = -\frac{u'}{\sqrt{1-u^2}}$
3. $\frac{d}{dx}[\arctan u] = \frac{u'}{1+u^2}$
4. $\frac{d}{dx}[\text{arccot} u] = -\frac{u'}{1+u^2}$
5. $\frac{d}{dx}[\text{arcsec} u] = \frac{u'}{|u|\sqrt{u^2-1}}$
6. $\frac{d}{dx}[\text{arccsc} u] = -\frac{u'}{|u|\sqrt{u^2-1}}$

Exponential and Logarithmic Functions

The exponential function $e^x$ is unique because its derivative is itself. Logarithms are the inverse of exponentials.

1. Exponential: $\frac{d}{dx}[e^u] = e^u \cdot u'$
2. General Exponential: $\frac{d}{dx}[a^u] = a^u (\ln a) \cdot u'$
3. Natural Logarithm: $\frac{d}{dx}[\ln u] = \frac{u'}{u}$
4. General Logarithm: $\frac{d}{dx}[\log_a u] = \frac{u'}{u \ln a}$

Hyperbolic Functions

Hyperbolic functions are defined using exponentials ($e^x$ and $e^{-x}$) and relate to hyperbolas similarly to how trig functions relate to circles.

1. $\sinh x = \frac{e^x - e^{-x}}{2}$
2. $\cosh x = \frac{e^x + e^{-x}}{2}$
3. $\tanh x = \frac{\sinh x}{\cosh x}$

Their derivatives parallel trigonometric functions, but with some sign differences:

1. $\frac{d}{dx}[\sinh u] = \cosh u \cdot u'$
2. $\frac{d}{dx}[\cosh u] = \sinh u \cdot u'$ (Positive!)
3. $\frac{d}{dx}[\tanh u] = \text{sech}^2 u \cdot u'$