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Mathematics and Statistics · Class 12 Commerce

Ch 3Differentiation — Class 12 Mathematics and Statistics, concept-first.

The derivative of a function measures its instantaneous rate of change — how fast the output responds to a small change in the input . Geometrically it is the slope of the tangent to the graph of at a point.

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Key concepts

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Rules of Differentiation

Computing every derivative directly from the limit definition ("from first principle") is correct but impractical for anything beyond the simplest functions.

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Chapter contents

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The Derivative — Recap and Standard Results

The derivative of a function measures its instantaneous rate of change — how fast the output responds to a small change in the input .

2

Rules of Differentiation — Sum, Product and Quotient

Most functions met in practice are built from simple pieces by addition, multiplication and division. Three rules handle these constructions.

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Derivative of Composite Functions — the Chain Rule

A composite function is a "function of a function", such as (a power applied to a polynomial) or (a logarithm applied to a polynomial).

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Derivatives of Inverse Functions

If is a differentiable function of , then (where the inverse exists and ) is a function of , and their derivatives are reciprocals: This inverse-function rule is invaluable when a relation is given as…

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Logarithmic Differentiation

Logarithmic differentiation takes the (natural) logarithm of both sides before differentiating. It is the right tool in two situations:

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Derivatives of Implicit Functions

A function is explicit when is written directly in terms of , as in . It is implicit when and are tangled together in an equation that is not (or cannot easily be) solved for , such as or .

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Derivatives of Parametric Functions

Sometimes both and are given separately in terms of a third variable (the parameter), as in . Such a pair is a parametric representation of a curve.

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Higher-Order Derivatives

The derivative of a function is itself a function of , so it can be differentiated again. Differentiating the first derivative gives the second derivative, written Differentiating once more gives the…

Exercises

+Show 5 questions5 questions
  1. Q12Differentiate $y = (x^2 + 2x)(3x - 1)$ with respect to $x$.Free
  2. Q13Find $\dfrac{dy}{dx}$ if $y = \sqrt{4x^2 + 1}$.Free
  3. Q14Differentiate $y = x^{\sin x}$ with respect to $x$.Preview
  4. Q15If $x^2 + xy + y^2 = 7$, find $\dfrac{dy}{dx}$, and its value at the point $(1, 2)$.Preview
  5. Q16If $y = e^{2x}$, find $\dfrac{d^2 y}{dx^2}$ and hence show that $\dfrac{d^2 y}{dx^2} = 4y$.Preview

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 21 questions21 questions
  1. Q1The slope of a tangent to the curve $y = 3x^2 - x + 1$ at $(1, 3)$ is ______. (a) 5 (b) $-5$ (c) $\dfrac{-1}{5}$ (d) $\dfrac{1}{5}$Preview
  2. Q2If $x = \sqrt{1 + u^2}$, $y = \log(1 + u^2)$, then find $\dfrac{dy}{dx}$.Preview
  3. Q3If $ax^2 + 2hxy + by^2 = 0$, then prove that $\dfrac{d^2y}{dx^2} = 0$.Preview
  4. Q4If $y = x \cdot \log x$ then $\frac{dy}{dx}$ = ______. (a) 1 (b) $\frac{1}{x}$ (c) $\log x$ (d) $1 + \log x$Preview
  5. Q5If $y = 2x^2 + 2^2 + a^2$, then $\frac{dy}{dx} = ?$ (a) $x$ (b) $4x$ (c) $2x$ (d) $-2x$ (e) $4x + 2a$ (f) $4x + 4$Preview
  6. Q6The derivative of $f(x) = a^x$, where a is constant is $x \cdot a^{x-1}$. (a) True (b) FalsePreview
  7. Q7If $y = (\log x)^2$ the $\frac{dy}{dx}$ = ______.Preview
  8. Q8If $x^5 \cdot y^7 = (x + y)^{12}$ then show that, $\frac{dy}{dx} = \frac{y}{x}$Preview
  9. Q9If $y = \log\left(\dfrac{e^x}{x^2}\right)$, then $\dfrac{dy}{dx} = ?$ (a) $\dfrac{2 - x}{x}$ (b) $\dfrac{x - 2}{x}$ (c) $\dfrac{e - x}{ex}$…Preview
  10. Q10Find $\dfrac{dy}{dx}$, if $x = e^{3t}$, $y = e^{(4t + 5)}$Preview
  11. Q11Find $\dfrac{dy}{dx}$ , if $y = x^{(e^x)}$Preview
  12. Q12Find $\frac{dy}{dx}$ if, $y = (x)^x + (a^x)$.Preview
  13. Q13If $x = \frac{4t}{1 + t^2}, \ y = 3\left(\frac{1 - t^2}{1 + t^2}\right)$ then show that $\frac{dy}{dx} = \frac{-9x}{4y}$.Preview
  14. Q14If $y = \sqrt[3]{a^2 + x^2}$ then $\frac{dy}{dx}$ = ______ (a) $\frac{2}{3}x(a^2 + x^2)^{-2/3}$ (b) $\frac{2}{3}x(a^2 + x^2)^{2/3}$ (c) $\fr…Preview
  15. Q15if $y = \log\left(\frac{e^x}{x^2}\right)$ then $\frac{dy}{dx}$ = ______. (a) $\frac{2 - x}{x}$ (b) $\frac{x - 2}{x}$ (c) $\frac{e - x}{e^x}$…Preview
  16. Q16State whether the following statement is true or false: If $y = 20 + 15x + x^2$ then $\frac{dx}{dy} = \frac{1}{15 + 2x}$Preview
  17. Q17If $y = e^{ax}$, then $x \cdot \frac{dy}{dx}$ = ______.Preview
  18. Q18If $e^x + e^y = e^{(x + y)}$, then show that $\frac{dy}{dx} = -e^{y - x}$.Preview
  19. Q19If $x = \log(1 + t^2)$ and $y = \log t$, then find $\dfrac{dy}{dx}$.Preview
  20. Q20If $x^2 y^k = (x + y)^{2 + k}$, then show that $\dfrac{dy}{dx} = \dfrac{y}{x}$.Preview
  21. Q21If $y = x^x + (7x - 1)^x$, then find $\dfrac{dy}{dx}$.Preview

More questions

+Show 11 questions11 questions
  1. Example 1Differentiate $y = (x^2 + 1)(x^3 + 2x)$ with respect to $x$ using the product rule.Free
  2. Example 2Find $\dfrac{dy}{dx}$ if $y = \dfrac{2x + 1}{x^2 + 3}$.Free
  3. Example 3Differentiate $y = (3x^2 + 5)^4$ with respect to $x$.Free
  4. Example 4Find $\dfrac{dy}{dx}$ if $y = \log(x^2 + 1)$.Preview
  5. Example 5If $x = y^3 + 2y$, find $\dfrac{dy}{dx}$, and evaluate it at the point where $y = 1$.Preview
  6. Example 6Using the inverse-function rule, show that $\dfrac{d}{dx}\left(\sin^{-1} x\right) = \dfrac{1}{\sqrt{1 - x^2}}$, and hence find its value at…Preview
  7. Example 7Differentiate $y = x^{x}$ with respect to $x$.Preview
  8. Example 8Find $\dfrac{dy}{dx}$ if $y = \dfrac{x^2 (2x+1)^3}{\sqrt{x+4}}$.Preview
  9. Example 9If $x^2 + y^2 = 25$, find $\dfrac{dy}{dx}$ by implicit differentiation, and its value at the point $(3, 4)$.Preview
  10. Example 10If $x = a t^2$ and $y = 2 a t$, find $\dfrac{dy}{dx}$ in terms of $t$.Preview
  11. Example 11If $y = x^4 - 3x^2$, find $\dfrac{d^2 y}{dx^2}$.Preview