The derivative of a function is a measure of how the function's output value changes as its input value changes. It is the limit of the average rate of change of the function over an interval as the interval becomes infinitesimally small.
How is the derivative written using notation?
The derivative is commonly denoted as f'(x) or dy/dx, where y = f(x).
What is the geometric meaning of a derivative?
The derivative represents the slope of the tangent line to the function at a given point.
Can a function be differentiable at a point where it is not continuous? Why or why not?
No, a function must be continuous at a point in order to be differentiable there. Differentiability implies continuity.
What is the derivative of a constant function?
The derivative of a constant function is 0.
Explain the power rule for differentiation.
The power rule states that if f(x) = x^n, then the derivative f'(x) = nx^(n-1), for any real number n.
What is the derivative of a linear function f(x) = ax + b?
The derivative of the linear function is simply the coefficient of x, which is 'a'.
Describe the product rule for differentiation.
The product rule is given by (fg)' = f'g + fg', meaning the derivative of a product of two functions is the derivative of the first times the second plus the first times the derivative of the second.
What is the chain rule and when is it used?
The chain rule is used to differentiate composite functions and is given by (f(g(x)))' = f'(g(x)) * g'(x).
What is the quotient rule?
The quotient rule is used when differentiating a division of two functions, given by (f/g)' = (f'g - fg')/g^2.
Explain why the function f(x) = |x| is not differentiable at x = 0.
The function f(x) = |x| is not differentiable at x = 0 because the graph has a sharp corner at this point, and thus the slope of the tangent line is not defined.
How do you find the derivative of the function f(x) = e^x?
The derivative of the function f(x) = e^x is e^x. The exponential function is its own derivative.
What is the derivative of sin(x)?
The derivative of sin(x) is cos(x).
What is implicit differentiation?
Implicit differentiation is a technique used to find the derivative of a function given in implicit form, where y is not isolated on one side of the equation.
What is the second derivative, and what does it represent?
The second derivative is the derivative of the derivative of a function. It represents the rate of change of the rate of change, or the curvature of the function's graph.