Theory (definitions and notation)
What is another term for "derivative" that relates to a graph?
Answer: slope of the tangent line
Use the limit definition of the derivative to find the derivative of f(x) = 2x
Answer: f'(x) = 2
What is the derivative of f(x) = x³?
Answer: f'(x) = 3x²
Find the derivative of f(x) = x²(x + 1)
Answer: f'(x) = 3x² + 2x
Find the derivative of f(x) = x / (x + 1)
Answer: f'(x) = 1 / (x + 1)²
The derivative of a function f(x) is often represented as f'(x). What is another common notation?
Answer: dy/dx
Find the derivative of f(x) = x + 1 using the limit definition
Answer: f'(x) = 1
Find the derivative of g(x) = 5x²
Answer: g'(x) = 10x
Differentiate g(x) = x(x³ - 2)
Answer: g'(x) = 4x³ - 2
Differentiate g(x) = 1 / x²
Answer: g'(x) = -2 / x³
The derivative is defined using a limit. What does this limit represent geometrically?
Answer: the slope of the tangent line
Use the limit definition to find the derivative of f(x) = 3
Answer: f'(x) = 0
Differentiate h(x) = x⁷
Answer: h'(x) = 7x⁶
What is the derivative of y = (x - 2)(x + 3)?
Answer: dy/dx = 2x + 1
What is the derivative of y = (x - 1) / x
Answer: dy/dx = 1 / x²
What does the expression lim (h→0) [f(x+h) - f(x)] / h represent?
Answer: the definition of the derivative
Find the derivative of f(x) = x² at x=1 using the limit definition
Answer: f'(1) = 2
What is the derivative of y = 2x⁴?
Answer: dy/dx = 8x³
Find dy/dx if y = 2x³(x - 1)
Answer: dy/dx = 8x³ - 6x²
Find dy/dx if y = x² / (x + 2)
Answer: dy/dx = (x² + 4x) / (x + 2)2
If a function's derivative is positive at a point, what does that indicate about the function at that point?
Answer: the function is increasing
Using limits, find the derivative of f(x) = 4x+2
Answer: f'(x) = 4
Find dy/dx if y = x10
Answer: dy/dx = 10x⁹
Differentiate h(x) = (x + 1)(x² - 1)
Answer: h'(x) = 3x² + 2x - 1
Differentiate h(x) = 3x / (x - 5)
Answer: h'(x) = -15 / (x - 5)²