Tangents, Normals, and Logarithmic Differentiation in Calculus

Tangent and Normal Lines to Implicit Curves

Introduction

In calculus, finding the equations of tangent and normal lines to curves—especially those defined implicitly—is a fundamental skill. These lines provide local linear approximations and geometric insight into the behavior of functions at specific points.

Implicit Differentiation

• Implicit differentiation is used when a function is defined by an equation involving both x and y, rather than explicitly as y = f(x).

• To find dy/dx (or y'), differentiate both sides of the equation with respect to x, treating y as a function of x.

Example: Find tangent and normal lines to $y^2 = x^2(2-x)$ at (1,1)

• Differentiating both sides: $2y y' = 2x(2-x) + x^2(-1)$

• Solve for $y'$: $y' = \frac{-x^2 + (2-x)2x}{2y}$

• At (1,1): $y' = \frac{-1^2 + (2-1)2(1)}{2(1)} = \frac{-1 + 2}{2} = \frac{1}{2}$

• Slope of tangent at (1,1) is $\frac{1}{2}$.

• Equation of tangent line: $y - 1 = \frac{1}{2}(x - 1)$

• Slope of normal line: $m_{normal} = -\frac{1}{m_{tangent}} = -2$

• Equation of normal line: $y - 1 = -2(x - 1)$

Example: Tangent and Normal to $x^2y^2 = (y+1)^2(4-y^2)$ at (0,-2)

• Differentiating both sides using the product and chain rules:

• At (0,-2), after substitution and simplification, $y' = 0$.

• Equation of tangent line: $y = -2$

• Slope of normal line: $m_{normal} = \frac{1}{0}$ (undefined, vertical line)

Logarithmic Differentiation

Introduction

Logarithmic differentiation is a powerful technique for differentiating complicated functions, especially those involving products, quotients, or variable exponents. It uses properties of logarithms to simplify the differentiation process.

Properties of Logarithms

Property	Equation
Product Rule	$\ln(ab) = \ln a + \ln b$
Quotient Rule	$\ln\left(\frac{a}{b}\right) = \ln a - \ln b$
Power Rule	$\ln a^b = b \ln a$

Steps in Logarithmic Differentiation

1. Take the natural logarithm of both sides of the equation $y = f(x)$.

2. Use the laws of logarithms to expand the expression.

3. Differentiate both sides implicitly with respect to $x$.

4. Solve for $y'$ and replace $y$ by $f(x)$.

Example: Differentiate $y = \frac{2x^3 (\tan x)^9}{x^2 + e^x}$

• Take logarithms: $\ln y = \ln(2x^3) + \ln((\tan x)^9) - \ln(x^2 + e^x)$

• Expand using properties: $\ln y = \ln 2 + 3 \ln x + 9 \ln(\tan x) - \ln(x^2 + e^x)$

• Differentiating both sides: $\frac{y'}{y} = \frac{3}{x} + \frac{9 \sec^2 x}{\tan x} - \frac{2x + e^x}{x^2 + e^x}$

• Multiply both sides by $y$ to solve for $y'$: $y' = y \left[ \frac{3}{x} + \frac{9 \sec^2 x}{\tan x} - \frac{2x + e^x}{x^2 + e^x} \right]$

• Substitute $y$ back: $y' = \frac{2x^3 (\tan x)^9}{x^2 + e^x} \left[ \frac{3}{x} + \frac{9 \sec^2 x}{\tan x} - \frac{2x + e^x}{x^2 + e^x} \right]$

Example: Differentiate $y = x^x$

• Take logarithms: $\ln y = x \ln x$

• Differentiating both sides: $\frac{y'}{y} = \ln x + 1$

• So, $y' = x^x (\ln x + 1)$

Example: Differentiate $y = (\sin x)^x$

• Take logarithms: $\ln y = x \ln(\sin x)$

• Differentiating both sides: $\frac{y'}{y} = \ln(\sin x) + x \cot x$

• So, $y' = (\sin x)^x [\ln(\sin x) + x \cot x]$

Example: Differentiate $y = \frac{(x^2+1)^2 e^{2x}}{\sin^5(3x)}$

• Take logarithms: $\ln y = 2 \ln(x^2+1) + 2x - 5 \ln(\sin(3x))$

• Differentiating both sides: $\frac{y'}{y} = \frac{4x}{x^2+1} + 2 - \frac{15 \cos(3x)}{\sin(3x)}$

• So, $y' = \frac{(x^2+1)^2 e^{2x}}{\sin^5(3x)} \left[ \frac{4x}{x^2+1} + 2 - \frac{15 \cos(3x)}{\sin(3x)} \right]$

Summary Table: Logarithmic Properties

Logarithm Law	Natural Logarithm	General Logarithm
Product	$\ln(ab) = \ln a + \ln b$	$\log_c(ab) = \log_c a + \log_c b$
Quotient	$\ln\left(\frac{a}{b}\right) = \ln a - \ln b$	$\log_c\left(\frac{a}{b}\right) = \log_c a - \log_c b$
Power	$\ln a^b = b \ln a$	$\log_c a^b = b \log_c a$

Key Points

• Tangent line at a point has slope equal to $y'$ at that point.

• Normal line at a point has slope $-1/y'$ (if $y' \neq 0$).

• Logarithmic differentiation simplifies differentiation of products, quotients, and variable exponents.

• Always use logarithm properties to expand before differentiating.

Applications

• Finding tangent and normal lines is essential in curve sketching, optimization, and physics (e.g., instantaneous velocity).

• Logarithmic differentiation is especially useful for functions like $y = x^x$, $y = (\sin x)^x$, and complicated rational or exponential expressions.

Additional info: The notes also provide graphical illustrations and step-by-step worked examples to reinforce the concepts.