Class 11 Maths and Statistics Part 1 Chapter 9: Differentiation

Short notes, key concepts, quick revision and practice MCQs with answers

Differentiation measures the rate of change of a function. The chapter defines the derivative as a limit, finds derivatives from first principles, lists the derivatives of standard functions and applies the sum, product and quotient rules, with simple applications such as rates of change and slopes of tangents.

Key concepts: Differentiation

Rate of change: The rate of change of y = f(x) with respect to x at a point is the derivative. Speed is the rate of change of distance with time.
Derivative at a point: f′(a) = lim(h→0) [f(a + h) − f(a)]/h, if the limit exists. Then f is differentiable at a.
First principle: dy/dx = lim(h→0) [f(x + h) − f(x)]/h. Used to prove the standard derivatives.
Differentiability and continuity: If f is differentiable at a, it is continuous at a. The converse is not true: |x| is continuous but not differentiable at x = 0.
Standard derivatives: d/dx(k) = 0, d/dx(xⁿ) = n xⁿ⁻¹, d/dx(√x) = 1/(2√x), d/dx(1/x) = −1/x², d/dx(eˣ) = eˣ, d/dx(aˣ) = aˣ log a, d/dx(log x) = 1/x.
Sum and constant rules: d/dx(u ± v) = du/dx ± dv/dx and d/dx(k u) = k du/dx.
Product and quotient rules: d/dx(uv) = u dv/dx + v du/dx. d/dx(u/v) = (v du/dx − u dv/dx)/v², v ≠ 0.
Applications: The derivative gives the slope of the tangent to a curve at a point and, in commerce, marginal quantities such as marginal cost and marginal revenue.

Quick revision: important points

d/dx(xⁿ) = n xⁿ⁻¹ holds for any real n.
d/dx(x log x) = 1 + log x.
d/dx(x eˣ) = eˣ(x + 1).
Slope of the tangent to y = f(x) at x = a is f′(a).
Every differentiable function is continuous.

Practice MCQs with answers

d/dx (x⁵) equals:
1. x⁴
2. 5x⁴
3. 5x⁶
4. x⁶/6
Show answer
Answer: B. 5x⁴
n xⁿ⁻¹ = 5x⁴.
d/dx (7) equals:
1. 7
2. 1
3. 0
4. 7x
Show answer
Answer: C. 0
The derivative of a constant is 0.
d/dx (eˣ) equals:
1. x eˣ⁻¹
2. eˣ
3. eˣ log e + 1
4. 1/eˣ
Show answer
Answer: B. eˣ
eˣ is its own derivative.
d/dx (log x) equals:
1. x
2. 1/x
3. log x / x
4. eˣ
Show answer
Answer: B. 1/x
Standard derivative.
d/dx (aˣ) equals:
1. x aˣ⁻¹
2. aˣ
3. aˣ log a
4. aˣ / log a
Show answer
Answer: C. aˣ log a
Standard derivative.

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