ilovepdf_merged (6).pdf). Content covers sections 28.1–28.x.Exponential and logarithmic functions appear throughout growth, decay, and inverse relationships. Their derivatives are especially clean in base e.
For y = [f(x)]^[g(x)], take ln: ln y = g ln f, then differentiate both sides (implicitly) to find y′. Useful for complicated products/powers.
d/dx e^(f(x)) = e^(f(x)) f′(x); d/dx ln f(x) = f′(x)/f(x) (f>0).
Most exam-important points from this chapter:
Start every answer with the key definition or standard form from Differentiation of Exponential and Logarithmic Functions.
Copy the formula, then substitute values; never jump to the number alone.
Check domain, quadrant, non-zero denominators, and applicability of the theorem.
Memorise and apply: and taking the limit as x
Memorise and apply: lim e
English questions from NIOS Mathematics (311) public / sample papers (Sample 2024, Apr 2024, Oct 2024, Apr 2025; board-style where scanned papers had no extractable text). Mapped exclusively to this chapter — no cross-chapter duplicates. Use Model Answer for the marking key and Explanation for working.
4 question(s) · Sources: Board-style
PYQ1. Differentiate y = eˣ + e^(−x).
Model Answer
1) eˣ − e^(−x)
Explanation
1) Chain rule on e^(−x) multiplies by −1.
PYQ2. Find d/dx (3ˣ).
Model Answer
1) 3ˣ ln 3
Explanation
1) d/dx aˣ = aˣ ln a with a=3.
PYQ3. If y = xˣ (x>0), find dy/dx.
Model Answer
1) xˣ (ln x + 1)
Explanation
1) ln y = x ln x ⇒ y′/y = ln x + 1 ⇒ y′ = xˣ(ln x + 1).
PYQ4. Differentiate y = e^(x²).
Model Answer
1) 2x e^(x²)
Explanation
1) Chain rule: e^(x²)·2x.
10 English exam-style problems for this chapter (NIOS Mathematics 311). Each question states the formula, gives full step-by-step working with the final answer, plus formal/easy explanations. English only.
Differentiate y=eˣ + e^(−x).
1) y′ = eˣ + e^(−x)·(−1) = eˣ − e^(−x).
Answer: eˣ − e^(−x)
Chain rule on e^(−x) produces a minus.
Second term: same e^(−x) times −1.
This derivative is 2 sinh x if that notation is used.
Sign on the second term.
Find d/dx (3ˣ).
1) 3ˣ ln 3.
Answer: 3ˣ ln 3
Standard exponential with base a>0, a≠1.
Keep 3ˣ and multiply by ln 3.
ln 3 is a constant.
Not 3ˣ / ln 3.
Differentiate y = ln(5x), x>0.
1) y=ln5 + ln x ⇒ y′=0 + 1/x = 1/x.
2) Or chain: (1/(5x))·5=1/x.
Answer: 1/x
ln(5x)=ln5+ln x; constant vanishes.
Same as derivative of ln x.
Domain x>0.
Not 1/(5x) final after chain cancel.
Differentiate y=e^(x²).
1) y′ = e^(x²) · 2x = 2x e^(x²).
Answer: 2x e^(x²)
Chain rule with f=x².
e to the power stays; multiply by 2x.
Order 2x e^(x²) is standard.
Do not write e^(2x).
If y = xˣ (x>0), find dy/dx.
1) ln y = x ln x.
2) y′/y = ln x + x·(1/x) = ln x + 1.
3) y′ = xˣ (ln x + 1).
Answer: xˣ (ln x + 1)
Logarithmic differentiation for variable base and exponent.
Take ln, differentiate, multiply by y.
Product rule on x ln x.
Do not treat as e^(x ln x) and forget product rule.
Differentiate y = x eˣ.
1) Product: 1·eˣ + x eˣ = eˣ(1+x).
Answer: eˣ(x + 1)
Product of x and eˣ.
eˣ + x eˣ factor eˣ.
Common factor eˣ.
Both terms needed.
Differentiate y = 5eˣ.
1) y′ = 5eˣ.
Answer: 5eˣ
Constant multiple rule with eˣ.
5 stays in front.
eˣ is unchanged by differentiation.
Check each algebraic step carefully.
Differentiate y = ln(x² + 1).
1) y′ = 1/(x²+1) · 2x = 2x/(x²+1).
Answer: 2x/(x² + 1)
Chain rule for ln of a function.
Derivative of inside over the inside.
Defined for all real x.
Check each algebraic step carefully.
Differentiate y = 2ˣ.
1) y′ = 2ˣ ln 2.
Answer: 2ˣ ln 2
Exponential base 2.
Keep 2ˣ; multiply by ln 2.
ln 2 is constant.
Check each algebraic step carefully.
If y = xˣ (x>0), show that y′ = xˣ(ln x + 1).
1) ln y = x ln x.
2) y′/y = ln x + 1.
3) y′ = xˣ (ln x + 1).
Answer: y′ = xˣ (ln x + 1)
Logarithmic differentiation for variable base and power.
Take ln, differentiate, multiply by y.
Product rule on x ln x.
Check each algebraic step carefully.