2.7 Derivatives and Integrals Foundations

Why This Section Matters

Derivatives and integrals complete the Functions and Calculus domain on Praxis 5165. ETS expects conceptual fluency: derivatives as slopes and instantaneous rates, integrals as signed area and accumulation, basic differentiation rules, and sign-chart reasoning for increasing/decreasing behavior and local extrema — not full AP Calculus BC depth.

Derivative as Instantaneous Rate

f'(x) measures how fast f(x) changes at x.

Basic Differentiation Rules

Power rule: if f(x) = x^n, then f'(x) = n · x^(n-1) (for n in the rationals tested on Praxis).

Examples:

• d/dx [3x²] = 6x
• d/dx [5x] = 5
• d/dx [7] = 0

Sum rule: derivative of a sum is the sum of derivatives.

Praxis 5165 may also expect d/dx [e^x] = e^x and d/dx [ln x] = 1/x for x > 0.

Sign Charts and Local Extrema

Worked Example: Minimum from f'

A sign chart shows f'(x) < 0 for x < 1 and f'(x) > 0 for x > 1.

f decreases then increases → local minimum at x = 1.

A local maximum requires f' changing from positive to negative.

Worked Example: Vertex via Derivative

For f(x) = -x² + 4x + 1:

f'(x) = -2x + 4. Set -2x + 4 = 0 → x = 2.

f''(x) = -2 < 0, so the parabola opens downward and x = 2 is a maximum — consistent with vertex formula.

Definite Integrals as Signed Area

∫_a^b f(x) dx represents signed area under f from x = a to x = b (area below the x-axis counts negative).

Worked Example: ∫₀² 3x dx

Antiderivative of 3x is (3/2)x².

[(3/2)(2)²] - 0 = (3/2)(4) = 6

Geometric check: triangle under y = 3x from 0 to 2 has base 2, height 6, area ½(2)(6) = 6.

Fundamental Theorem (Conceptual)

If F'(x) = f(x), then ∫_a^b f(x) dx = F(b) - F(a).

This links antiderivatives to net accumulation — total change when f is a rate.

Example: if v(t) is velocity, ∫_0^5 v(t) dt gives displacement over [0, 5] (with sign).

Applications for Teacher Candidates

• Position / velocity / acceleration links
• Marginal interpretations in simple economic models
• Connecting f' sign to student sketches of increasing/decreasing intervals
• Explaining why a critical point is not automatically a maximum

Second Derivative (Light Touch)

f''(x) > 0 suggests concave up; f''(x) < 0 suggests concave down. For f(x) = -x² + 4x + 1, f''(x) = -2 confirms concave down everywhere.

Common Praxis Traps

• Confusing f(x) with f'(x) on sign charts
• Reporting only F(a) instead of F(b) - F(a)
• Calling every critical point a minimum without sign change in f'
• Forgetting definite integrals measure signed area

Section Takeaways

Use derivatives for slope and monotonicity, definite integrals for accumulated area, and sign charts for extrema. Finish with timed /practice/praxis-math calculus-tagged items before moving to geometry.

Product and Chain Rules (When Needed)

Product rule: if h(x) = f(x)g(x), then h'(x) = f'(x)g(x) + f(x)g'(x).

Example: d/dx [x² · sin x] = 2x sin x + x² cos x.

Chain rule: if h(x) = f(g(x)), then h'(x) = f'(g(x)) · g'(x).

Example: d/dx [(3x + 1)⁴] = 4(3x + 1)³ · 3 = 12(3x + 1)³.

Praxis 5165 uses these at moderate depth — often recognizing setup rather than lengthy computation.

Antiderivatives and +C

The family of antiderivatives of f(x) = 2x is x² + C. Only definite integrals with bounds produce a single numeric value without C.

∫ 2x dx = x² + C (indefinite) ∫₀¹ 2x dx = 1 (definite)

Area Between Curves (Conceptual)

Area between f(x) and g(x) on [a, b] when f(x) ≥ g(x) is ∫_a^b [f(x) - g(x)] dx. Even if full setup is rare, recognizing subtraction of lower from upper function is tested.

Connecting Derivative and Integral

Derivatives and integrals are inverse operations in the Fundamental Theorem sense. If F'(x) = 3x², then ∫ 3x² dx = x³ + C = F(x) + C.

A teaching item might ask which statement best explains why antiderivatives are not unique — because any vertical shift + C has the same derivative.

Optimization Preview

To maximize A(x) = -x² + 4x + 1, set A'(x) = 0 → x = 2, then verify maximum via sign chart or A''(x) < 0. Optimization links derivative zeros to real modeling questions on fencing, revenue, or area under constraints.

Riemann Sum Intuition (Conceptual)

Definite integrals can be approximated by summing rectangle areas Σ f(x_i) Δx. As partition width shrinks, the sum approaches ∫ f(x) dx. Praxis may show a left- or right-Riemann diagram and ask which integral it approximates — count subintervals and heights carefully.

Units in Applied Integrals

If r(t) is a rate in gallons per minute, then ∫_0^10 r(t) dt has units gallons — total accumulated quantity. Tracking units validates whether an integral or derivative setup matches a word problem.

When f' = 0 but No Extremum

If f'(x) = 0 but the derivative sign does not change (e.g., f(x) = x³ at x = 0), there is an inflection point, not a local max or min. Sign-chart analysis must show an actual sign change for extrema conclusions.