Differentiation Review for Exam

Set of flashcards reviewing key concepts in differentiation, including derivatives, rules of differentiation, and geometric interpretations.

The derivative of a function f at a number a, denoted by f'(a), is __.

m = f'(a) = lim (f(x)-f(a))/(x-a) as x approaches a.

The slope of the tangent line to the curve y = f(x) at the point (a, f(a)) is __.

the derivative f'(a).

To find an equation of the tangent line, you use the formula __.

y - y₁ = m(x - x₁).

The __ rule is used for differentiating products of two functions.

Product Rule.

The __ rule is used for differentiating quotients of two functions.

Quotient Rule.

If f is differentiable at x=a, then f is __ at x=a.

continuous at x=a.

The __ rule states that the derivative of a composition of functions is the product of the derivative of the outer function and the derivative of the inner function.

Chain Rule.

The derivative of the function f(x) = sin(x) is __.

cos(x).

The derivative of the function f(x) = e^x is __.

e^x.

The second derivative of f is denoted as __.

f''(x) or f(2)(x).

The derivative of y = ln(x + 1) is __.

1/(x + 1).

The geometric interpretation of f'(a) is the __ of the tangent to f at (a, f(a)).

slope.

The position function s(t) represents the __ of an object over time.

location.

The derivative of y = x^n, according to the Power Rule, is __.

n*x^(n-1).

The __ differentiation implies that if f is continuous at x=a, it is not necessarily differentiable at x=a.

Theorem of Continuity.

The height function of a ball thrown in the air can be modeled as h(t) = __.

40t - 16t^2.

If f is a one-to-one function, then its inverse f^-1 has a derivative given by __.

1/f'(f^-1(x)).

The derivative of the function f(x) = cos(x) is __.

-sin(x).

The __ rule states that d[c*f(x)]/dx = c * d[f(x)]/dx for a constant c.

Constant Multiple Rule.

To find the instantaneous rate of change of a function at a point, you evaluate the __ at that point.

derivative.

If f(x) = tan(x), the derivative f'(x) = __.

sec^2(x).

If f(x) is not differentiable at a point, it could be due to __ at that point.

discontinuity, corner, cusp, or vertical tangent.

The limit definition of the derivative involves a limit as __ approaches zero.

h.