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Mathematics and Statistics · Class 11 Commerce

Ch 9Differentiation — Class 11 Mathematics and Statistics, concept-first.

Differentiation measures how fast one quantity changes with respect to another. If , the derivative of with respect to tells us the instantaneous rate of change of for a change in — geometrically, the slope of the tangent to the curve at a point.

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Key concepts

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Derivative from First Principles

The derivative of a function at a point is formally defined as , provided this limit exists, in which case is called differentiable at .

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Chapter contents

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1

The Derivative as a Rate of Change

Differentiation measures how fast one quantity changes with respect to another. If , the derivative of with respect to tells us the instantaneous rate of change of for a change in — geometrically, the…

2

The Derivative from First Principles

The formal definition of the derivative — the one all the shortcut rules are ultimately built from — is the limit of the average rate of change: provided this limit exists.

3

Derivatives of Standard Functions

Differentiating every function from first principles would be slow, so the results for common functions are established once and memorised as a table of standard derivatives.

4

Rules of Differentiation

Most functions are combinations of the standard ones, and four algebraic rules let us differentiate any such combination. Let and be differentiable, and a constant.

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The Chain Rule (Composite Functions)

A composite function is a "function of a function", such as or or — an outer function applied to an inner expression. The chain rule differentiates it.

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The Second-Order Derivative

The derivative is itself a function of , so it can be differentiated again. The result is the second-order derivative (or simply second derivative): It is found in two steps: differentiate once to get…

Exercises

+Show 6 questions6 questions
  1. Q12Differentiate $f(x) = 3x^{2}$ from first principles.Free
  2. Q13Differentiate $y = 7x^{5} + 4\sqrt{x} - \dfrac{3}{x}$.Free
  3. Q14Differentiate $y = (x^{2} + 1)(x^{3} + 2)$ using the product rule.Preview
  4. Q15Differentiate $y = \dfrac{2x + 3}{x^{2} + 1}$ using the quotient rule.Preview
  5. Q16Differentiate $y = \sqrt{4x^{2} + 1}$ using the chain rule.Preview
  6. Q17If $y = \sin x$, find $\dfrac{d^{2}y}{dx^{2}}$.Preview

More questions

+Show 11 questions11 questions
  1. Example 1Differentiate $f(x) = x^{2}$ from first principles.Free
  2. Example 2Differentiate $f(x) = \dfrac{1}{x}$ from first principles.Free
  3. Example 3Differentiate $f(x) = \sqrt{x}$ from first principles.Free
  4. Example 4Differentiate $y = 5x^{4} - 3x^{3} + 2x - 7$.Preview
  5. Example 5Differentiate $y = x^{2} e^{x}$ using the product rule.Preview
  6. Example 6Differentiate $y = \dfrac{x^{2} + 1}{x - 1}$ using the quotient rule.Preview
  7. Example 7Differentiate $y = (3x^{2} + 5)^{4}$ using the chain rule.Preview
  8. Example 8Differentiate $y = \sin(x^{2})$.Preview
  9. Example 9Differentiate $y = e^{3x + 2}$.Preview
  10. Example 10If $y = x^{4}$, find the second-order derivative $\dfrac{d^{2}y}{dx^{2}}$.Preview
  11. Example 11For the curve $y = x^{3} - 2x$, find the slope of the tangent at $x = 2$.Preview