The derivative f'(x) is defined as: — lim_{h→0} [f(x+h)−f(x)]/h.The derivative is the limit of the difference quotient as h → 0: f'(x) = lim_{h→0}[f(x+h)−f(x)]/h.
Geometrically, f'(a) represents: — The slope of the tangent line to f at x=a.The derivative at a point equals the slope of the tangent line at that point.
The difference quotient [f(x+h)−f(x)]/h represents: — The average rate of change over interval h.The difference quotient is the average rate of change over a small interval h. As h→0, it becomes the instantaneous rate.
Using the definition, f'(x) for f(x) = x²: — 2x.lim_{h→0}[(x+h)²−x²]/h = lim[(2xh+h²)/h] = lim[2x+h] = 2x.
Using the limit definition, the derivative of f(x) = c (constant) is: — 0.f'(x) = lim[c−c]/h = lim[0/h] = 0. The derivative of a constant is always zero.
The derivative of f(x) = 5x using the limit definition: — 5.lim[(5(x+h)−5x)/h] = lim[5h/h] = lim[5] = 5.
The derivative of f(x) = x³ (using definition or power rule) is: — 3x².d/dx[x³] = 3x². (Or verify via limit: lim[(x+h)³−x³]/h = lim[3x²+3xh+h²] = 3x².)
The notation f'(x), dy/dx, and Df(x) all mean: — The derivative of f with respect to x.These are three equivalent notations for the derivative: Lagrange notation f'(x), Leibniz notation dy/dx, and operator notation Df(x).
Using the limit definition, find f'(2) for f(x) = x²: — 4.f'(x) = 2x. f'(2) = 2(2) = 4.
The slope of the tangent line to y = x² at x = 3 is: — 6.dy/dx = 2x. At x = 3: slope = 2(3) = 6.
The second derivative f''(x) is: — The derivative of the derivative (f').The second derivative is f''(x) = d/dx[f'(x)] — the derivative of the first derivative.
If f'(x) > 0 on an interval, then f is: — Increasing.Positive derivative means the function is increasing (slope is positive).
The power rule: d/dx[xⁿ] = ? — nxⁿ⁻¹.d/dx[xⁿ] = nxⁿ⁻¹. Multiply by the exponent and reduce the power by 1.
d/dx[x⁵] = ? — 5x⁴.d/dx[x⁵] = 5x⁴. Power rule: bring down 5, reduce power to 4.
d/dx[eˣ] = ? — eˣ.The exponential function eˣ is its own derivative: d/dx[eˣ] = eˣ.
d/dx[ln x] = ? — 1/x.d/dx[ln x] = 1/x. This is one of the most important derivative formulas.
In implicit differentiation, we differentiate both sides with respect to: — x.We differentiate both sides with respect to x. When a y-term appears, the chain rule introduces dy/dx.
When differentiating y² implicitly with respect to x: — 2y·(dy/dx).d/dx[y²] = 2y·(dy/dx). Chain rule: y is a function of x, so we multiply by its derivative dy/dx.
In related rates, we differentiate all variables with respect to: — Time t.Related rates problems involve quantities that all change with time. We differentiate with respect to t.
Which strategy is FIRST in solving related rates problems? — Draw a diagram.First draw a diagram and label variables. Then write a relating equation, THEN differentiate, THEN substitute.
A 10-foot ladder slides. x is the base from wall, y is the top height. The constraint is: — x² + y² = 100.Pythagorean theorem: x² + y² = 10² = 100.
The key to related rates is recognizing which quantities are: — Changing with time.Related rates problems involve quantities that all change with time (or another variable). Identifying which quantities vary is the first conceptual step.
A critical point of f occurs where: — f'(x) = 0 or f'(x) is undefined.Critical points occur where f'(x) = 0 or where f'(x) doesn't exist. These are candidates for local extrema.
If f'(x) > 0 on (a,b), then f is: — Increasing on (a,b).Positive first derivative ↔ increasing function.
If f'(x) < 0 on (a,b), then f is: — Decreasing on (a,b).Negative first derivative ↔ decreasing function.
The tangent line to y = f(x) at point (a, f(a)) is: — y − f(a) = f'(a)(x − a).Tangent line equation: y − f(a) = f'(a)(x − a). Uses point-slope form with derivative as slope.