logarithmetic differentiation, Review, and Related Rate

Logarithmic differentiation

Application in handling functions of the form $f (x)^{g (x)}$ or $\frac{f_{1} (x) f_{2} (x) \dots f_{n} (x)}{g_{1} (x) g_{2} (x) \dots g_{m} (x)}$
Two-Step Process:
- Step 1: Take the natural logarithm $\ln$ of the given equation
- Step 2: Apply implicit differentiation to find the derivative

Review technique of differentiation

table of derivatives of basic functions

Function	Derivative
Constant (c)	$\frac{d}{d x} (c) = 0$
Power Rule	$\frac{d}{d x} (x^{n}) = n x^{(n - 1)}$
Exponential (e^x)	$\frac{d}{d x} (e^{x}) = e^{x}$
Natural Logarithm (ln(x))	$\frac{d}{d x} (\ln (x)) = \frac{1}{x}$
Sine (sin(x))	$\frac{d}{d x} (\sin (x)) = \cos (x)$
Cosine (cos(x))	$\frac{d}{d x} (\cos (x)) = - \sin (x)$
Tangent (tan(x))	$\frac{d}{d x} (\tan (x)) = \sec^{2} (x)$
Natural Log (log_b(x))	$\frac{d}{d x} (\log_{b} (x)) = \frac{1}{x \ln (b)}$
Inverse Sine (arcsin(x))	$\frac{d}{d x} (\arcsin (x)) = \frac{1}{\sqrt{1 - x^{2}}}$
Inverse Cosine (arccos(x))	$\frac{d}{d x} (\arccos (x)) = - \frac{1}{\sqrt{1 - x^{2}}}$
Inverse Tangent (arctan(x))	$\frac{d}{d x} (\arctan (x)) = \frac{1}{1 + x^{2}}$
Inverse Cotangent (arccot(x))	$\frac{d}{d x} (\cot^{- 1} (x)) = - \frac{1}{1 + x^{2}}$
Inverse Secant (arcsec(x))	$\frac{d}{d x} (\sec^{- 1} (x)) = \frac{1}{x \sqrt{x^{2} - 1}}$
Inverse Cosecant (arccsc(x))	$\frac{d}{d x} (\csc^{- 1} (x)) = - \frac{1}{x \sqrt{x^{2} - 1}}$

differential rules:
1. Sum/Difference Rule: $\frac{d}{d x} (f (x) \pm g (x)) = \frac{d}{d x} (f (x)) \pm \frac{d}{d x} (g (x))$ .
2. Product Rule: $\frac{d}{d x} (f (x) \cdot g (x)) = f^{'} (x) \cdot g (x) + f (x) \cdot g^{'} (x)$ .
3. Quotient Rule: $\frac{d}{d x} (\frac{f (x)}{g (x)}) = \frac{f^{'} (x) \cdot g (x) - f (x) \cdot g^{'} (x)}{(g (x))^{2}}$ .
4. Chain Rule: $\frac{d}{d x} (f (g (x))) = f^{'} (g (x)) \cdot g^{'} (x)$ .
implicit differentiation (inverse functions and logarithmetic differentiation):
1. inverse functions: $y = f^{- 1} (x)$ implies $f (y) = x$ .
2. logarithmetic differentiation: $y = f (x)^{g (x)}$ implies $\ln (y) = g (x) \ln (f (x))$ .

Investigate scenarios where we are given one rate of change and are tasked with determining another rate of change. Understand the concept of relative rates and how to calculate them.

Logarithmic differentiation

Review technique of differentiation

Related rate