Solving Calculus Problems On Differentiation

 

What is Differentiation?

Differentiation is a fundamental operation in calculus that measures the rate at which a function changes relative to its input variable. Geometrically, finding the derivative of a function at a specific point gives the slope of the tangent line to the curve at that exact point.

If y = f(x), the derivative is denoted as ^dy⁄_dx or f'(x), and it represents the instantaneous rate of change of y with respect to x.

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The First 10 Rules of Differentiation

1. The Constant Rule: The derivative of any constant is zero.
   ^d⁄_dx(c) = 0
2. The Power Rule: Bring the power to the front and subtract one.
   ^d⁄_dx(xⁿ) = n · x^n-1
3. The Constant Multiple Rule: Factor out the constant before differentiating.
   ^d⁄_dx(c · f(x)) = c · f'(x)
4. The Sum Rule: Differentiate terms separately.
   ^d⁄_dx(f(x) + g(x)) = f'(x) + g'(x)
5. The Difference Rule: Differentiate terms separately.
   ^d⁄_dx(f(x) - g(x)) = f'(x) - g'(x)
6. The Product Rule: First times derivative of the second, plus second times derivative of the first.
   ^d⁄_dx(f(x) · g(x)) = f(x)g'(x) + g(x)f'(x)
7. The Quotient Rule: "Low d-high minus high d-low, over the square of what's below."
   ^d⁄_dx( ^f(x)⁄_g(x) ) = g(x)f'(x) - f(x)g'(x) ⁄ _[g(x)]²
8. The Chain Rule: Derivative of the outer function multiplied by the derivative of the inner function.
   ^d⁄_dx(f(g(x))) = f'(g(x)) · g'(x)
9. The Exponential Rule (Base e): The natural exponential function is its own derivative.
   ^d⁄_dx(e^x) = e^x
10. The Logarithmic Rule (Natural Log): The derivative of ln(x) is 1/x.
    ^d⁄_dx(ln(x)) = ¹⁄_x

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Practice Quiz (10 Objective Questions)

Question 1 (Constant Rule): What is the derivative of f(x) = 157?

• A) 157
• B) 1
• C) 0
• D) 157x

View Answer & Solution

Correct Answer: C

Solution: According to the Constant Rule, the rate of change of any constant value is always zero because it does not vary with x. Therefore, ^d⁄_dx(157) = 0.

Question 2 (Power Rule): Find the derivative of f(x) = x⁷.

• A) 7x⁷
• B) 7x⁶
• C) 6x⁷
• D) ^x8⁄₈

View Answer & Solution

Correct Answer: B

Solution: Apply the Power Rule, ^d⁄_dx(xⁿ) = n · x^n-1. Here, n = 7. Bringing the 7 to the front and subtracting 1 from the power gives 7x^7-1 = 7x⁶.

Question 3 (Constant Multiple Rule): Differentiate f(x) = 4x⁵.

• A) 20x⁴
• B) 4x⁴
• C) 20x⁵
• D) 5x⁴

View Answer & Solution

Correct Answer: A

Solution: Keep the constant coefficient 4 aside and differentiate x⁵ using the Power Rule (5x⁴). Then multiply them back together: 4 · (5x⁴) = 20x⁴.

Question 4 (Sum Rule): Find ^dy⁄_dx if y = x³ + x².

• A) 5x⁴
• B) 3x² + x
• C) 3x² + 2x
• D) 6x

View Answer & Solution

Correct Answer: C

Solution: The Sum Rule states you can differentiate each term independently. The derivative of x³ is 3x² and the derivative of x² is 2x. Adding them together yields 3x² + 2x.

Question 5 (Difference Rule): Find the derivative of f(x) = 6x² - 2x.

• A) 12x - 2
• B) 12x
• C) 6x - 2
• D) 12x² - 2

View Answer & Solution

Correct Answer: A

Solution: Differentiate each term independently across the subtraction sign. ^d⁄_dx(6x²) = 12x and ^d⁄_dx(2x) = 2. This leaves us with 12x - 2.

Question 6 (Product Rule): Differentiate y = x² · e^x.

• A) 2x · e^x
• B) x² · e^x + 2x · e^x
• C) x² + e^x
• D) 2x² · e^x

View Answer & Solution

Correct Answer: B

Solution: Let f(x) = x² → f'(x) = 2x, and g(x) = e^x → g'(x) = e^x. Applying the Product Rule f(x)g'(x) + g(x)f'(x), we obtain (x²)(e^x) + (e^x)(2x), which is x²e^x + 2xe^x.

Question 7 (Quotient Rule): Find the derivative of f(x) = ln(x) ⁄ x.

• A) 1 ⁄ x²
• B) (1 - ln(x)) ⁄ x
• C) (1 - ln(x)) ⁄ x²
• D) (ln(x) - 1) ⁄ x²

View Answer & Solution

Correct Answer: C

Solution: Let the top function be u = ln(x) → u' = 1/x and the bottom function be v = x → v' = 1.
Applying the Quotient Rule (vu' - uv') / v² gives:
[x(1/x) - ln(x)(1)] ⁄ x² = (1 - ln(x)) ⁄ x².

Question 8 (Chain Rule): Differentiate f(x) = (2x + 3)⁴.

• A) 4(2x + 3)³
• B) 8(2x + 3)³
• C) 8(2x + 3)⁴
• D) 2(2x + 3)³

View Answer & Solution

Correct Answer: B

Solution: Differentiate the outside power expression first, leaving the inside expression intact: 4(2x+3)³. Then multiply by the derivative of the inside expression, ^d⁄_dx(2x+3) = 2. This results in 4(2x+3)³ · 2 = 8(2x+3)³.

Question 9 (Exponential Rule): Find the derivative of f(x) = e^5x.

• A) e^5x
• B) 5e^x
• C) 5e^5x
• D) (1/5)e^5x

View Answer & Solution

Correct Answer: C

Solution: The derivative of e^u is e^u · u'. Here, u = 5x and its derivative is 5. Therefore, the overall derivative is e^5x · 5 = 5e^5x.

Question 10 (Logarithmic Rule): Find the derivative of f(x) = ln(4x).

• A) 1 ⁄ x
• B) 1 ⁄ 4x
• C) 4 ⁄ x
• D) 4ln(x)

View Answer & Solution

Correct Answer: A

Solution: Using the chain rule version of the log rule: ^d⁄_dx(ln(u)) = (1/u) · u'. Substituting u = 4x and u' = 4, we get (1/4x) · 4 = 4/4x = 1/x.