AP Calculus AB - Differentiation and Applications

Front

What is the limit definition of the derivative of a function f at x = a?

Back

The derivative f'(a) is defined as the limit of the difference quotient: lim_{h->0} [f(a+h) - f(a)] / h, provided the limit exists. Alternatively, it can be defined as lim_{x->a} [f(x) - f(a)] / (x - a). This represents the instantaneous rate of change of f at x = a.

Front

If f'(c) > 0, what does this indicate about the function f at x = c?

Back

A positive derivative at x = c indicates that the function f is increasing at that point. The slope of the tangent line to the graph of f at the point (c, f(c)) is positive.

Front

State the Power Rule for differentiation.

Back

If f(x) = x^n, where n is a real number, then the derivative is f'(x) = nx^(n-1). For example, the derivative of x^5 is 5x^4, and the derivative of 1/x (which is x^-1) is -1/x^2.

Front

How do you find the equation of the tangent line to the graph of f at x = a?

Back

To find the tangent line: 1) Calculate the point of tangency: (a, f(a)). 2) Find the slope of the tangent line: m = f'(a). 3) Use the point-slope form of a line: y - f(a) = f'(a)(x - a).

Front

What is the Product Rule for differentiation?

Back

If f(x) = g(x) * h(x), then f'(x) = g'(x) * h(x) + g(x) * h'(x). In words: the derivative of a product is the derivative of the first times the second, plus the first times the derivative of the second.