Implicit Differentiation

Implicit Differentiation

Differentiation Introduction

• Differentiation is the process of finding the derivative of a function.
• This marks the rate at which a quantity is changing at a given point.
• The derivative represents the instantaneous rate of change of the function.

Basic Rules

• Power Rule: (d/dx[x^n] = nx^{n-1})
• Sum Rule: (d/dx[f(x) + g(x)] = f’(x) + g’(x))
• Difference Rule: (d/dx[f(x) - g(x)] = f’(x) - g’(x))

Derivatives of Common Functions

• The derivative of (ln(x)) is (1/x).
• The derivative of (e^x) is (e^x).
• The derivative of (a^x) is (a^x \cdot ln(a)).

Chain Rule

• The Chain Rule is used to differentiate complex functions.
• Formula: (d/dx[f(g(x))] = f’(g(x)) \cdot g’(x))

Trigonometric Functions

• The derivative of (sin(x)) is (cos(x)).
• The derivative of (cos(x)) is (-sin(x)).
• The derivative of (tan(x)) is (sec^2(x)).

Product and Quotient Rules

• Product Rule: (d/dx[f(x) \cdot g(x)] = f’(x) \cdot g(x) + f(x) \cdot g’(x))
• Quotient Rule: (d/dx[f(x) / g(x)] = [f’(x) \cdot g(x) - f(x) \cdot g’(x)]/[g(x)]^2)

Stationary Points

• A point is stationary if its derivative is zero.
• Maxima and minima are types of stationary points.
• The second derivative test can be used to classify stationary points.

Convex and Concave Functions

• If a function’s second derivative is positive, the function is concave up or convex.
• If it’s negative, the function is concave down or concave.
• Points where concavity changes are inflection points.

Parametric Equations

• When a function is expressed in terms of more than one variable, it’s called parametric.
• To differentiate with respect to one variable, we differentiate implicitly and treat the variable as a constant.

Implicit Differentiation

• Implicit Differentiation is needed when a function is not explicitly written as (y = f(x)).
• It involves differentiating each term separately and then simplifying.
• This is often essential when dealing with curves and shapes defined by implicit functions.