Calculus: Difference Quotient and Tangent Lines

Difference Quotient

• The difference quotient is used to calculate the slope of a tangent line to a curve at a point.
• There are two forms:
  • f(x + Δx) - f(x) / Δx
  • f(x) - f(a) / x - a

Finding the Slope of a Tangent Line

• The slope of a line tangent to a curve at a point represents the instantaneous rate of change of the curve at that point.
• The tangent line only touches the curve at the given point, unlike a secant line which intersects the curve at two points.
• As the distance between two points on the curve approaches zero, the slope of the secant line approaches the slope of the tangent line.
• The limit as Δx approaches zero of the difference quotient f(x + Δx) - f(x) / Δx represents the slope of the tangent line at a given point.

Δx and the Difference Quotient

• Δx represents the change in the x values or the distance between two x values.
• f(x + Δx) represents the function evaluated at x + Δx, which is the x value plus the distance between the x values.

Derivative

• The derivative is a function that represents the slope of its parent function at a particular point.
• It can be found by evaluating the difference quotient at a specific value of x or by finding the limit of the difference quotient as Δx approaches zero.

Instantaneous Rate of Change

• The instantaneous rate of change is the rate of change of a function at a specific point in time.
• The tangent line represents the instantaneous rate of change at a point.

Evaluating the Difference Quotient

• To evaluate the difference quotient at a specific x value, substitute that x value into the function.
• To find the instantaneous rate of change at a specific x value, evaluate the difference quotient at that x value or find the limit of the difference quotient as Δx approaches zero.

The Derivative

• The derivative of a function represents the slope of the original graph at any given point.
• The derivative can be found using the difference quotient:
  • limit as Delta X approaches zero of (f(x + Delta X) - f(x)) / Delta X
• This equation gives you the slope of the tangent line at a given point.
• To find the slope at x = 2, for example, you would plug in 2 for x in the derivative function.

Differentiability

• A function is differentiable if and only if it has a tangent line at every point.
• A function is not differentiable if it is discontinuous.
• A function is not differentiable if it has a corner.
  • A corner represents a point on the graph where the slope approaches two different values from the left and from the right.
  • A corner does not have to be a 90-degree angle.
• A function is not differentiable if it has a cusp - a special kind of corner.
  • A cusp involves a graph where the one-sided limits of the difference quotient approach negative infinity and positive infinity, respectively.
  • Since the one-sided limits do not exist, the limit of the difference quotient does not exist, and therefore the function is not differentiable.
• A function is not differentiable if it has a vertical tangent line.
  • While the one-sided limits of the difference quotient may be equal to each other, they still equal positive infinity.
  • Since the limit does not exist, the function is not differentiable.
• A function is locally linear, which means that if you zoom in far enough on the graph, it will appear linear.

Derivative Rules

• The derivative of a constant, such as y = 4, is always zero because the slope of a horizontal line is always zero. This is true for any constant value.
• The derivative of a first degree function, such as y = ax (where a is a constant), is always equal to a.
• The derivative of a function in the form of ax^n (where a and n are constants) is equal to a * n * x^(n-1).
• The derivative of a constant multiple of a function, such as f(x) = 4x², is equal to the constant multiple outside of the derivative operator.

Summary of Differentiability

• A function is differentiable if it is continuous and has no corners or cusps.
• A function is continuous if it is differentiable.
• The intermediate value theorem applies to derivatives: If there are two points on the derivative function, then the derivative function must achieve every point between those two points.
• The intermediate value theorem is based on the fact that the derivative function is continuous because the original function was differentiable.
• When we solve for the derivative of a function, we find a pattern.
• These patterns form the basis for the derivative rules, which encompass a universal set of rules for taking derivatives.

Derivative Rules

• Power Rule When two like bases are multiplied, add their exponents. This applies to functions with invisible exponents of one, resulting in an *x* raised to the power of (*n* + 1), where *n* is the original exponent.
• Constant Multiple Rule The constant multiple of a function can be factored out of the derivative operation.
• Product Rule Used for finding the derivative of a function *h(x)* that can be represented as the product of two functions, *f(x)* and *g(x)*:
  • Formula: *h'(x) = f'(x) * g(x) + g'(x) * f(x)*
• Quotient Rule: Used for finding the derivative of a function *h(x)* that can be represented as the quotient of two functions, *f(x)* and *g(x)*:
  • Formula: *h'(x) = (g(x) * f'(x) - f(x) * g'(x)) / (g(x))^2*
• Mnemonic for Quotient Rule: "Low D High - High D Low, over Low squared."

Trigonometric Derivatives

• Sine: The derivative of *sin(x)* is *cos(x)*
• Cosine: The derivative of *cos(x)* is *-sin(x)*
• Tangent: The derivative of *tan(x)* is *(sec(x))^2*
• Cosecant: The derivative of *csc(x)* is *-csc(x) * cot(x)*
• Secant: The derivative of *sec(x)* is *sec(x) * tan(x)*
• Cotangent: The derivative of *cot(x)* is *-csc(x)^2*
• Memorization Tip: Derivatives of trig functions starting with 'c' (cosine, cosecant, cotangent) have negative signs.

Inverse Trigonometric Derivatives

• Inverse Sine: The derivative of *sin^-1(x)* is *1 / (√(1 - x^2)) *
• Inverse Cosine: The derivative of *cos^-1(x)* is *-1 / (√(1 - x^2)) *
• Inverse Tangent: The derivative of *tan^-1(x)* is *1 / (1 + x^2) *
• Inverse Cosecant: The derivative of *csc^-1(x)* is *-1 / (|x| * √(x^2 - 1)) *
• Inverse Secant: The derivative of *sec^-1(x)* is *1 / (|x| * √(x^2 - 1)) *
• Inverse Cotangent: The derivative of *cot^-1(x)* is *-1 / (x^2 + 1). *

Exponential and Logarithmic Derivatives

• Logarithmic Function: The derivative of *log_a(x)* is *1 / (x * ln(a)) *
• Natural Logarithm: The derivative of *ln(x)* is *1 / x*
• Exponential Function: The derivative of *a^x* is *a^x * ln(a)*
• Natural Exponential Function: The derivative of *e^x* is *e^x*
• Note: Common mistake: The derivative of a function like *2^x* is *2^x * ln(2)*, not a product rule problem because *ln(2)* is a constant.

This quiz covers the concept of the difference quotient and its application in finding the slope of a tangent line to a curve. Understand how to use the difference quotient formulas and the significance of Δx in calculus. Test your knowledge on calculating instantaneous rates of change using these principles.