Advanced Techniques for Computing Derivatives

Advanced Techniques for Computing Derivatives

Once the basic differentiation rules and the basic computation methods are mastered, we need to learn some advanced techniques to handle more complicated differentiation problems.

Higher-Order Derivatives

Second Derivative

Physical meaning: acceleration is the derivative of velocity, that is, the second derivative of displacement.

Examples:

• $f(x) = x^2$, $f'(x) = 2x$, $f''(x) = 2$
• $f(x) = \sin x$, $f'(x) = \cos x$, $f''(x) = -\sin x$

n-th Derivative

Higher-order derivatives of common functions:

• $(x^n)^{(n)} = n!$
• $(e^x)^{(n)} = e^x$
• $(\sin x)^{(n)} = \sin(x + \frac{n\pi}{2})$
• $(\cos x)^{(n)} = \cos(x + \frac{n\pi}{2})$

Implicit Differentiation

The Basic Method

Steps:

1. Differentiate both sides of the equation simultaneously
2. Solve for $\frac{dy}{dx}$, treating it as an unknown
3. Rearrange to get an expression for $\frac{dy}{dx}$

Example: Find the derivative of $x^2 + y^2 = 1$.

Solution:

1. Differentiate both sides: $2x + 2y\frac{dy}{dx} = 0$
2. Rearrange: $2y\frac{dy}{dx} = -2x$
3. Obtain: $\frac{dy}{dx} = -\frac{x}{y}$

Complicated Implicit Functions

Example: Find the derivative of $x^3 + y^3 = 3xy$.

Solution:

1. Differentiate both sides: $3x^2 + 3y^2\frac{dy}{dx} = 3y + 3x\frac{dy}{dx}$
2. Rearrange: $3y^2\frac{dy}{dx} - 3x\frac{dy}{dx} = 3y - 3x^2$
3. Factor out the common term: $(3y^2 - 3x)\frac{dy}{dx} = 3y - 3x^2$
4. Obtain: $\frac{dy}{dx} = \frac{y - x^2}{y^2 - x}$

Differentiation of Parametric Equations

The Basic Formula

Explanation: the derivative of a parametric curve equals the derivative of $y$ with respect to the parameter divided by the derivative of $x$ with respect to the parameter.

Example: Find the derivative of the parametric equations $x = t^2$, $y = t^3$.

Solution:

• $\frac{dx}{dt} = 2t$
• $\frac{dy}{dt} = 3t^2$
• $\frac{dy}{dx} = \frac{3t^2}{2t} = \frac{3t}{2}$

Higher-Order Derivatives

Second derivative: $\frac{d^2y}{dx^2} = \frac{d}{dx}(\frac{dy}{dx}) = \frac{d}{dt}(\frac{dy}{dx}) \cdot \frac{dt}{dx}$

Example: Find the second derivative of the parametric equations $x = \cos t$, $y = \sin t$.

Solution:

1. First derivative: $\frac{dy}{dx} = \frac{\cos t}{-\sin t} = -\cot t$
2. Second derivative: $\frac{d^2y}{dx^2} = \frac{d}{dt}(-\cot t) \cdot \frac{1}{-\sin t} = \csc^2 t \cdot \frac{1}{-\sin t} = -\csc^3 t$

Logarithmic Differentiation

When to Use It

Logarithmic differentiation is useful in the following situations:

• complicated power-exponential functions
• products or quotients of several functions
• expressions that contain both exponentials and logarithms

The Basic Steps

1. Take the logarithm of the function
2. Differentiate both sides of the equation
3. Solve for $\frac{dy}{dx}$

Examples

Example 1: Find the derivative of $y = x^x$.

Solution:

1. Take the logarithm: $\ln y = x \ln x$
2. Differentiate: $\frac{1}{y}\frac{dy}{dx} = \ln x + 1$
3. Solve: $\frac{dy}{dx} = y(\ln x + 1) = x^x(\ln x + 1)$

Example 2: Find the derivative of $y = \frac{(x+1)^2(x-2)^3}{(x+3)^4}$.

Solution:

1. Take the logarithm: $\ln y = 2\ln(x+1) + 3\ln(x-2) - 4\ln(x+3)$
2. Differentiate: $\frac{1}{y}\frac{dy}{dx} = \frac{2}{x+1} + \frac{3}{x-2} - \frac{4}{x+3}$
3. Solve: $\frac{dy}{dx} = y\left(\frac{2}{x+1} + \frac{3}{x-2} - \frac{4}{x+3}\right)$

Differentiation of Piecewise Functions

The Basic Method

Method:

1. Differentiate each piece separately
2. Check differentiability at the breakpoints

Example

Example: $f(x) = \begin{cases} x^2, & x \geq 0 \\ -x^2, & x < 0 \end{cases}$

Solution:

• When $x > 0$: $f'(x) = 2x$
• When $x < 0$: $f'(x) = -2x$
• At $x = 0$: $f'(0) = 0$

Advanced Computation Techniques

1. Simplification

Technique: simplify the function before differentiating it

Example:

• Original function: $f(x) = \frac{x^2 - 1}{x - 1}$
• Simplified: $f(x) = x + 1$ (when $x \neq 1$)
• Derivative: $f'(x) = 1$

2. Exploiting Symmetry

Technique: use the symmetry of the function to simplify the computation

Example: For an even function $f(x) = f(-x)$:

• $f'(x) = -f'(-x)$
• At $x = 0$, $f'(0) = 0$

For an odd function $f(x) = -f(-x)$:

• $f'(x) = f'(-x)$
• the derivative is an even function

3. Decomposing Composite Functions

Technique: decompose a complicated function into a composition of simple ones

Example: $f(x) = \sin(e^{x^2})$

Decomposition:

• $u = x^2$
• $v = e^u$
• $w = \sin v$
• $f(x) = w$

Differentiation:

• $\frac{dw}{dv} = \cos v = \cos(e^{x^2})$
• $\frac{dv}{du} = e^u = e^{x^2}$
• $\frac{du}{dx} = 2x$
• $f'(x) = \cos(e^{x^2}) \cdot e^{x^2} \cdot 2x = 2xe^{x^2}\cos(e^{x^2})$

Common Errors and Precautions

1. Mistakes in Implicit Differentiation

Mistake: forgetting to apply the chain rule when differentiating $y$ Correct: when differentiating a term that contains $y$, multiply by $\frac{dy}{dx}$

Example:

• Mistake: $(y^2)' = 2y$
• Correct: $(y^2)' = 2y\frac{dy}{dx}$

2. Mistakes in Differentiation of Parametric Equations

Mistake: differentiating $x$ and $y$ directly Correct: differentiate with respect to the parameter first, then use the formula

3. Mistakes in Logarithmic Differentiation

Mistake: forgetting to multiply by the original function at the end Correct: $\frac{dy}{dx} = y \cdot \frac{d}{dx}(\ln y)$

---

Exercises

Exercise 1

Find the derivative of the implicit function $x^2 + xy + y^2 = 1$.

Exercise 2

Find the derivative of the parametric equations $x = t^3$, $y = t^2$.

Exercise 3

Find the derivative of $y = x^{\sin x}$ using logarithmic differentiation.

Exercise 4

Find the derivative of $f(x) = \begin{cases} x^2, & x \geq 0 \\ -x^2, & x < 0 \end{cases}$.

---

Summary

Symbols Used in This Article