Differentiation and Applications

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Differentiation — Key Facts

Differentiation finds the rate of change of a function with respect to its variable. If y = f(x), then dy/dx (or f’(x)) gives the instantaneous rate of change of y with respect to x.

Basic Derivatives:

Function	Derivative
$x^n$	$nx^{n-1}$
$\sin x$	$\cos x$
$\cos x$	$-\sin x$
$\tan x$	$\sec^2 x$
$e^x$	$e^x$
$\ln x$	$\frac{1}{x}$
$a^x$	$a^x \ln a$

Rules of Differentiation:

• Sum/Difference: $\frac{d}{dx}(f \pm g) = f’ \pm g’$
• Product: $\frac{d}{dx}(fg) = f’g + fg’$
• Quotient: $\frac{d}{dx}\left(\frac{f}{g}\right) = \frac{f’g - fg’}{g^2}$
• Chain Rule: $\frac{d}{dx}[f(g(x))] = f’(g(x)) \cdot g’(x)$

⚡ ECAT Exam Tip: For chain rule: derivative of “function of function” equals derivative of outside × derivative of inside. Example: d/dx(sin³x) = 3sin²x · cosx.

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding…

Applications of Derivatives:

1. Equation of Tangent and Normal:

At point (x₁, y₁) on curve y = f(x):

• Slope of tangent: $m = \left(\frac{dy}{dx}\right)_{(x_1,y_1)}$
• Equation of tangent: $y - y_1 = m(x - x_1)$
• Equation of normal: $y - y_1 = -\frac{1}{m}(x - x_1)$

2. Increasing and Decreasing Functions:

• f is increasing on (a, b) if f’(x) > 0 for all x in (a, b)
• f is decreasing on (a, b) if f’(x) < 0 for all x in (a, b)

3. Maxima and Minima:

Critical points when f’(x) = 0 or f’(x) doesn’t exist.

First Derivative Test:

• If f’ changes from + to - at x = c: local maximum at c
• If f’ changes from - to + at x = c: local minimum at c
• If f’ doesn’t change sign: point of inflection

Second Derivative Test:

• If f’(c) = 0 and f”(c) < 0: local maximum
• If f’(c) = 0 and f”(c) > 0: local minimum
• If f”(c) = 0 or doesn’t exist: test inconclusive, use first derivative test

4. Rate of Change:

If two quantities x and y are related by y = f(x), then: $$\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt}$$

5. Mean Value Theorem:

If f is continuous on [a, b] and differentiable on (a, b), then there exists c in (a, b) such that: $$f’(c) = \frac{f(b) - f(a)}{b - a}$$

⚡ ECAT Exam Tip: For optimisation problems: identify the quantity to optimise, express it in terms of one variable, differentiate and set equal to zero, verify it’s a maximum/minimum using second derivative test.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Advanced Differentiation:

1. Implicit Differentiation:

When y cannot be expressed explicitly as a function of x:

• Differentiate both sides with respect to x
• Treat y as a function of x: d/dx(y) = dy/dx
• Solve for dy/dx

Example: $x^2 + y^2 = 25$ Differentiating: $2x + 2y \frac{dy}{dx} = 0$ $\frac{dy}{dx} = -\frac{x}{y}$

2. Logarithmic Differentiation:

For functions of the form y = f(x)^g(x) or products of many functions:

• Take ln of both sides
• Differentiate using chain rule

Example: $y = x^x$ (x > 0) $\ln y = x \ln x$ Differentiating: $\frac{1}{y} \frac{dy}{dx} = \ln x + 1$ $\frac{dy}{dx} = x^x(1 + \ln x)$

3. Derivatives of Inverse Functions:

If y = f⁻¹(x), then: $$\frac{dy}{dx} = \frac{1}{f’(y)} = \frac{1}{f’(f^{-1}(x))}$$

Function	Derivative
$\sin^{-1}x$	$\frac{1}{\sqrt{1-x^2}}$
$\cos^{-1}x$	$-\frac{1}{\sqrt{1-x^2}}$
$\tan^{-1}x$	$\frac{1}{1+x^2}$
$e^x$	$e^x$
$\ln x$	$\frac{1}{x}$

4. Second Order Derivatives:

• $\frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right) = f”(x)$
• For motion: if s = f(t), then v = ds/dt, a = dv/dt = d²s/dt²

5. Leibniz Rule (nth Derivative of Product):

$$\frac{d^n}{dx^n}(uv) = \sum_{k=0}^{n} \binom{n}{k} \frac{d^{n-k}u}{dx^{n-k}} \cdot \frac{d^k v}{dx^k}$$

6. L’Hôpital’s Rule:

For indeterminate forms 0/0 or ∞/∞: $$\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f’(x)}{g’(x)}$$

(Can be applied repeatedly if needed)

Example: $\lim_{x \to 0} \frac{\sin x}{x} = \lim_{x \to 0} \frac{\cos x}{1} = 1$

7. Curvature:

Radius of curvature: $\rho = \frac{[1 + (y’)^2]^{3/2}}{|y”|}$

8. Tangent and Normal in 3D:

For curve r(t) = (x(t), y(t), z(t)):

• Tangent vector: r’(t)
• Unit tangent: $\hat{T} = \frac{r’}{|r’|}$
• Normal vector: $\hat{N} = \frac{\hat{T}’}{|\hat{T}’|}$
• Binormal: $\hat{B} = \hat{T} \times \hat{N}$

⚡ ECAT 2024 Analysis: Questions on chain rule, maxima/minima, and rate of change appear frequently. Logarithmic differentiation is particularly important for functions like x^sinx or (x²+1)^x. For practical problems, always check that your answer makes physical sense — a maximum volume should be positive, etc.

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