04 The Derivative Free Online FlashCards

What does a derivative measure?

A derivative measures how rapidly a function changes as its input changes; geometrically, it is the slope of the curve at a point.

What is the average rate of change?

The average rate of change from x = a to x = b is [f(b) − f(a)]/(b − a), the slope of the secant line.

State the limit definition of f′(a).

f′(a) = limₕ→₀ [f(a + h) − f(a)]/h, provided the limit exists. This is the instantaneous rate of change at a.

What is a difference quotient?

The difference quotient is [f(a + h) − f(a)]/h. It is undefined at h = 0, but its limit as h approaches zero may exist.

List four common derivative notations.

Equivalent derivative notations include f′(x), y′, dy/dx, and d/dx[f(x)].

What is the tangent-line equation at x = a?

The tangent line at x = a is y − f(a) = f′(a)(x − a). Its slope is f′(a) and it passes through (a, f(a)).

What does it mean for f to be differentiable at a?

Differentiability at a means that f′(a) exists. A derivative can fail at a discontinuity, corner, cusp, or vertical tangent.

What is the relationship between differentiability and continuity?

Differentiability at a implies continuity at a, but continuity does not imply differentiability. For example, |x| is continuous but not differentiable at 0.

State the power rule.

The power rule is d/dx[xⁿ] = nxⁿ⁻¹ for any real number n.

What is the product rule?

The product rule is d/dx[f(x)g(x)] = f′(x)g(x) + f(x)g′(x).

State the chain rule.

The chain rule is d/dx[f(g(x))] = f′(g(x))g′(x): differentiate the outer function, then multiply by the inner derivative.

Differentiate 4x⁵ − 3x² + 7x − 9.

For f(x) = 4x⁵ − 3x² + 7x − 9, term-by-term differentiation gives f′(x) = 20x⁴ − 6x + 7.