Derivatives 1

Define the difference quotient of a function at a point.

For f: A ⊆ ℝ → ℝ and x₀ ∈ int(A), the difference quotient is [f(x₀+h) − f(x₀)]/h, where h ≠ 0 and x₀+h ∈ A. It measures the average change in f per unit change in x.

Define derivability of a function at an interior point.

A function f: A ⊆ ℝ → ℝ is derivable at x₀ ∈ int(A) if the limit f′(x₀) = lim as h→0 of [f(x₀+h) − f(x₀)]/h exists and is finite.

Explain the geometric meaning of the difference quotient.

It is the slope of the secant line through (x₀, f(x₀)) and (x₀+h, f(x₀+h)).

State the geometric meaning of the derivative and the equation of the tangent line.

The derivative f′(x₀) is the slope of the tangent line at (x₀, f(x₀)). Its equation is y − f(x₀) = f′(x₀)(x − x₀).

What conditions must the limit of the difference quotient satisfy for a function to be derivable?

The two-sided limit must exist and be finite. An infinite limit does not give a derivative under the definition used in these slides.

Define a vertical inflection point.

A vertical inflection point is a point where the graph changes concavity and has a vertical tangent. The difference quotient tends to +∞ or −∞, so no finite derivative exists. Examples at 0: f(x) = ∛x and f(x) = −∛x.

Define a horizontal inflection point.

A horizontal inflection point is a point where the graph changes concavity and has a horizontal tangent, so f′(x₀) = 0. Example: f(x) = x³ at 0. A zero derivative alone does not establish an inflection point.

Define the right derivative at a point.

The right derivative is f′₊(x₀) = lim as h→0⁺ of [f(x₀+h) − f(x₀)]/h, provided this limit exists and is finite.

Define the left derivative at a point.

The left derivative is f′₋(x₀) = lim as h→0⁻ of [f(x₀+h) − f(x₀)]/h, provided this limit exists and is finite.

State the theorem relating derivability to the existence and equality of one-sided derivatives.

For f: (a,b) → ℝ and x₀ ∈ (a,b), f is derivable at x₀ if and only if its right and left derivatives both exist, are finite, and are equal. In this case, f′(x₀) = f′₊(x₀) = f′₋(x₀).

Define a corner.

A corner is a point where the left and right derivatives are both finite but unequal. The two-sided derivative does not exist. Example: f(x) = |x| at 0, where f′₋(0) = −1 and f′₊(0) = 1.

Define a cusp.

A cusp is a point where the one-sided difference quotients tend to infinities of opposite signs. No finite derivative exists. Example: f(x) = √|x| at 0, with left limit −∞ and right limit +∞.

Define the derivative function and its domain.

For f: (a,b) → ℝ, let D ⊆ (a,b) be the set of points where f is derivable. The derivative function f′: D → ℝ assigns to every x ∈ D the derivative f′(x).

State the derivatives of a constant function and an affine function.

If f(x) = k, then f′(x) = 0. If f(x) = mx+q, then f′(x) = m, where k, m, and q are real constants.

State the power rule and its domain restrictions.

For f(x) = x^α, f′(x) = αx^(α−1). For every real α, the rule holds on x > 0. Its applicability at negative x or at 0 depends on the exponent and must be checked separately.

State the derivatives of exponential functions.

For a > 0, (a^x)′ = a^x·ln(a). In particular, (e^x)′ = e^x. These formulas hold for every real x.

State the derivatives of the natural logarithm and a logarithm with an arbitrary base, including domain and base restrictions.

For x > 0, (ln x)′ = 1/x. For x > 0, a > 0, and a ≠ 1, (logₐ x)′ = 1/[x·ln(a)] = (logₐ e)/x.

State the derivatives of sine and cosine.

(sin x)′ = cos x and (cos x)′ = −sin x, for every real x, with angles measured in radians.

State the derivative of tangent and where the formula applies.

(tan x)′ = 1/cos²(x) = 1+tan²(x), where cos(x) ≠ 0; equivalently, x ≠ π/2+kπ for k ∈ ℤ. Angles are measured in radians.

State the derivative of arctangent.

(arctan x)′ = 1/(1+x²), for every real x.

State the theorem on the derivative of a sum, including its hypotheses.

If f and g are derivable at x, then f+g is derivable at x and (f+g)′(x) = f′(x) + g′(x).

State the rule for differentiating a linear combination, including its hypotheses.

If f and g are derivable at x and α, β ∈ ℝ are constants, then αf+βg is derivable at x and (αf+βg)′(x) = αf′(x) + βg′(x).

State the product rule, including its hypotheses.

If f and g are derivable at x, then fg is derivable at x and (fg)′(x) = f′(x)g(x) + f(x)g′(x).

State the rule for differentiating a product of three functions.

If f, g, and h are derivable at x, then (fgh)′(x) = f′(x)g(x)h(x) + f(x)g′(x)h(x) + f(x)g(x)h′(x).

State the quotient rule, including its hypotheses.

If f and g are derivable at x and g(x) ≠ 0, then f/g is derivable at x and (f/g)′(x) = [f′(x)g(x) − f(x)g′(x)]/[g(x)]².

State the chain rule, including its hypotheses.

Let f: (a,b) → ℝ and g: (c,d) → ℝ, with Im(f) ⊆ (c,d). If f is derivable at x and g is derivable at f(x), then g∘f is derivable at x and (g∘f)′(x) = g′(f(x))·f′(x).

State the chain rule for a composition of three functions.

If the compositions are defined and f, g, and h are derivable at the relevant points, then (h∘g∘f)′(x) = h′(g(f(x)))·g′(f(x))·f′(x).

State the theorem on the derivative of an inverse function, including its hypotheses.

Let f be continuous and injective on an open interval, derivable at x₀, with f′(x₀) ≠ 0. Then f⁻¹ is derivable at y₀ = f(x₀), and (f⁻¹)′(y₀) = 1/f′(x₀). Equivalently, (f⁻¹)′(y) = 1/f′(f⁻¹(y)).

Explain the geometric relationship between a function and its inverse, including their tangent slopes.

The graphs of f and f⁻¹ are reflections in the line y = x. Corresponding points are (x₀,y₀) and (y₀,x₀). When the inverse-derivative theorem applies, their tangent slopes are reciprocals.