Calculus Derivatives Practice Problems Set #1

Derivative Functions and Their Definitions

• For the function f(x) = 4x³ - 3x² + 2x + 5, the derivative is f '(x) = 12x² - 6x + 2.
• For f(x) = 4x⁵ - 5x⁴, the derivative is f '(x) = 20x⁴ - 20x³.
• A constant function f(x) = 21 has a derivative of f '(x) = 0.
• The derivative for the product f(x) = (x² + 1)(x - 1) is f '(x) = 3x² - 2x + 1.
• The function f(x) = x⁻³ has the derivative f '(x) = -3x⁻⁴.
• For the quadratic function f(x) = -x² + 3, the derivative is f '(x) = -2x.
• The derivative of the polynomial f(x) = 3x⁷ - 7x³ + 21x² is f '(x) = 21x⁶ - 21x² + 42x.
• The product of linear functions f(x) = (2x + 5)(3x - 2) has a derivative of f '(x) = 12x + 11.
• The reciprocal function f(x) = -2x⁻² results in the derivative f '(x) = 4x⁻³.
• For the product f(x) = (x + 7)(x - 5), the derivative is f '(x) = 2x + 2.

Trigonometric Derivatives

• For the product of trigonometric functions g(x) = cos(x)tan(x), the derivative is g'(x) = cos(x).
• The derivative of the cotangent function J(x) = cot(x) is J'(x) = -csc²(x).
• The derivative of secant is represented by p(x) = sec(x) yielding dp/dx = sec(x)tan(x).

Logarithmic Derivative

• The derivative of the natural logarithm is f(x) = ln(x) with f'(x) = 1/x.

Rules for Derivatives

• The Product Rule states that the derivative of a product d/dx (f(x)g(x)) is f '(x) g(x) + f(x) g'(x).
• The Quotient Rule for derivatives, d/dx (f(x)/g(x)), is given by [f '(x) g(x) - f(x) g'(x)] / [g(x)]².
• The Power Rule indicates that for d/dx( x^n ), the derivative is nx^(n -1).

Understanding Derivatives

• A derivative provides the slope of the tangent line to the graph of f(x) at a specific point x = a.
• Derivatives represent the instantaneous rate of change of f(x) at x = a.
• They can be used to determine the rate of change at any point on the function f(x) through the function f '(x).