Application of Derivatives Formulas

All key formulas grouped by subtopic. Each one has a quick reminder and common mistakes to watch for.

10 formulas · 7 subtopics

Rate of Change

\frac{dy}{dx} = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}

💡 dy/dx gives the instantaneous rate of change of y with respect to x. For related rates, use chain rule: dy/dt = (dy/dx)(dx/dt).

Equation of Tangent

y - y_1 = f'(x_1)(x - x_1)

💡 The slope of the tangent at (x1, y1) is f'(x1). If f'(x1) = 0, the tangent is horizontal: y = y1.

Equation of Normal

y - y_1 = -\frac{1}{f'(x_1)}(x - x_1), \quad f'(x_1) \neq 0

💡 Normal is perpendicular to the tangent. Its slope = -1/f'(x1). If f'(x1) = 0, the normal is vertical: x = x1.

Length of Tangent, Normal, Subtangent, Subnormal

\text{Subtangent} = \frac{y_1}{f'(x_1)}, \quad \text{Subnormal} = y_1 \cdot f'(x_1)

💡 Length of tangent = |y1|sqrt(1 + 1/[f'(x1)]^2). Length of normal = |y1|sqrt(1 + [f'(x1)]^2). These are measured from the point to the x-axis intercept along tangent/normal.

Condition for Increasing Function

f'(x) > 0 \text{ on } (a, b) \Rightarrow f \text{ is strictly increasing on } [a, b]

💡 f'(x) >= 0 (with equality only at isolated points) also gives strictly increasing. Check open interval for derivative sign.

Condition for Decreasing Function

f'(x) < 0 \text{ on } (a, b) \Rightarrow f \text{ is strictly decreasing on } [a, b]

💡 Similar to increasing: f'(x) <= 0 (with equality only at isolated points) also gives strictly decreasing.

First Derivative Test

f'(x) \text{ changes sign: } +\to- \Rightarrow \text{local max}, \quad -\to+ \Rightarrow \text{local min}

💡 Find critical points where f'(x) = 0 or f'(x) does not exist. Check sign of f'(x) on either side of each critical point.

Second Derivative Test

f'(c) = 0 \text{ and } f''(c) < 0 \Rightarrow \text{local max}; \quad f''(c) > 0 \Rightarrow \text{local min}

💡 If f''(c) = 0, the test is inconclusive. Fall back to the first derivative test in that case.

⚠ Applying the second derivative test when f''(c) = 0 and concluding it is a point of inflection without further analysis.

Global Max/Min on Closed Interval [a, b]

\text{Global max/min of } f \text{ on } [a,b] = \max/\min\{f(a),\, f(c_1),\, f(c_2),\, \ldots,\, f(b)\}

💡 Evaluate f at all critical points inside (a, b) AND at both endpoints a and b. The largest value is the global max, smallest is the global min.

Linear Approximation

#10

f(x + \Delta x) \approx f(x) + f'(x) \cdot \Delta x

💡 Used to approximate values like sqrt(4.01), (3.98)^(1/2), etc. Choose x as the nearest value where f is easy to compute.

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Limits, Continuity & Differentiability

Differential Equations

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