University:
California State University, Northridge
Course:
MATH 351 | Differential Equations
Academic year:

2024
Views:

97

Pages:

3
Author:

Javon Neal

Question 1 Derivatives of other trigonometric functions. Verify the following derivative formulas using the Quotient Rule. d/dx (sec x) = sec x tan x Answer Step 1 The secant function in terms of cosine can be written as sec x = 1 / cos x. The derivative of cos x is –sin x. The quotient rule states: d/dx [f(x)/g(x)] = [g(x)f'(x) – f(x)g'(x)] / [g(x)]². Step 2 To compute d/dx (sec x): d/dx [1 / cos x] = [0 – (–sin x)] / (cos x)². Simplify: d/dx (sec x) = sin x / cos² x = (1 / cos x) * (sin x / cos x) = sec x tan x. Result: The derivative formula is verified. Question 2 What is the derivative of the sigmoid function: σ(x) = 1 / (1 + e⁻ˣ) Answer The derivative is: d/dx [σ(x)] = σ(x) (1 – σ(x)) Step 1: Rewrite: σ(x) = 1 / (1 + e⁻ˣ). Differentiate using chain rule: d/dx [σ(x)] = d/dx [1 / (1 + e⁻ˣ)]. Simplify: d/dx [σ(x)] = (–e⁻ˣ) / (1 + e⁻ˣ)². Factor: d/dx [σ(x)] = σ(x) (1 – σ(x)). Alternative Solution: Using quotient rule and simplifying, we find the same result: f'(x) = σ(x) (1 – σ(x)). Question 3 Use formulas (1) and (2) and the power rule to find the derivatives of f(x) = x² / 3. Answer Step 1 Given function: f(x) = x² / 3. Step 2 Differentiate using the power rule: d/dx (xⁿ) = n xⁿ⁻¹. Step 3 d/dx [x² / 3] = (2x) / 3. Result: The derivative of f(x) is: d/dx [f(x)] = 2x / 3. Question 4 Find the derivatives of the following functions using chain rule: f(x) = (4x² – 5x)³. Answer Step 1 Let y = (4x² – 5x)³. Step 2 Using chain rule: d/dx [y] = 3(4x² – 5x)² d/dx (4x² – 5x). Differentiate inner term: d/dx (4x² – 5x) = 8x – 5. Combine: d/dx [y] = 3(4x² – 5x)² (8x – 5). Result: The derivative is: d/dx [f(x)] = 3(8x – 5)(4x² – 5x)². Question 5 Differentiate the following function using product rule: f(x) = (3x – 9)(x – 6). Answer Step 1 The product rule states: d/dx [f(x)g(x)] = f'(x)g(x) + f(x)g'(x). Step 2 Given: f(x) = (3x – 9)(x – 6). Differentiate: f'(x) = (3)(x – 6) + (3x – 9)(1). Simplify: f'(x) = 3(x – 6) + (3x – 9). f'(x) = 3x – 18 + 3x – 9. f'(x) = 6x – 27. Result: The derivative of the function is: f'(x) = 6x – 27.