📈 \( \frac{dy}{dx} \)
📈 Chapter 12: Derivatives | NEB Mathematics Notes Class 12
Based on Latest Syllabus 2080 | Differentiation Rules & Applications
✅ Updated according to latest syllabus of 2080 | Complete notes with 15 solved problems
📈 Derivatives — This chapter covers the concept of derivatives, differentiation rules, hyperbolic functions, higher-order derivatives, logarithmic differentiation, and applications. The notes have been updated according to the latest syllabus of 2080. Now you don’t need to go anywhere searching for the notes of this chapter because we are here to serve you.
📚 Exercise
📖 Definition of Derivative
If \(y = f(x)\) is a function of \(x\), the derivative of \(y\) with respect to \(x\) is denoted by \(\frac{dy}{dx}\) or \(f'(x)\). It is defined as:
\[
f'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) – f(x)}{\Delta x}
\]
The derivative measures how a function’s output changes as its input changes. For example, if \(f(x)\) represents position, then \(f'(x)\) gives velocity.
📐 Derivative Rules
Power Rule: \(\frac{d}{dx}(x^n) = nx^{n-1}\)
Sum/Difference Rule: \(\frac{d}{dx}[u \pm v] = u’ \pm v’\)
Product Rule: \(\frac{d}{dx}(uv) = u’v + uv’\)
Quotient Rule: \(\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u’v – uv’}{v^2}\)
Chain Rule: \(\frac{dy}{dx} = f'(g(x)) \cdot g'(x)\)
🌀 Derivatives of Hyperbolic Functions
\(\frac{d}{dx}(\sinh x) = \cosh x\)
\(\frac{d}{dx}(\cosh x) = \sinh x\)
\(\frac{d}{dx}(\tanh x) = \text{sech}^2 x\)
\(\frac{d}{dx}(\text{sech } x) = -\text{sech } x \tanh x\)
\(\frac{d}{dx}(\text{coth } x) = -\text{csch}^2 x\)
\(\frac{d}{dx}(\text{csch } x) = -\text{csch } x \coth x\)
📊 Higher Order Derivatives
The second derivative is the derivative of the derivative: \(\frac{d^2y}{dx^2}\) or \(f”(x)\). It measures the rate of change of the rate of change (acceleration).
\[ f”(x) = \frac{d}{dx}\left(f'(x)\right) \]
\[ f”(x) = \frac{d}{dx}\left(f'(x)\right) \]
🔢 Logarithmic Differentiation
For functions of the form \(y = [f(x)]^{g(x)}\), take natural logarithm of both sides:
\[
\ln y = g(x) \ln f(x)
\]
Then differentiate both sides with respect to \(x\):
\[
\frac{1}{y}\frac{dy}{dx} = g'(x)\ln f(x) + g(x)\frac{f'(x)}{f(x)}
\]
📝 Solved Problems (15 Important Questions)
Q1 Differentiate \(y = x^5 + 3x^3 – 2x + 7\) with respect to x.
\(\frac{dy}{dx} = 5x^4 + 9x^2 – 2\)
✅ \(\frac{dy}{dx} = 5x^4 + 9x^2 – 2\)
Q2 Differentiate \(y = (2x+3)(x^2-4)\) using product rule.
Let \(u = 2x+3\), \(v = x^2-4\), \(u’=2\), \(v’=2x\)
\(\frac{dy}{dx} = u’v + uv’ = 2(x^2-4) + (2x+3)(2x) = 2x^2 – 8 + 4x^2 + 6x = 6x^2 + 6x – 8\)
\(\frac{dy}{dx} = u’v + uv’ = 2(x^2-4) + (2x+3)(2x) = 2x^2 – 8 + 4x^2 + 6x = 6x^2 + 6x – 8\)
✅ \(\frac{dy}{dx} = 6x^2 + 6x – 8\)
Q3 Differentiate \(y = \frac{x^2+1}{x-1}\) using quotient rule.
Let \(u = x^2+1\), \(v = x-1\), \(u’=2x\), \(v’=1\)
\(\frac{dy}{dx} = \frac{u’v – uv’}{v^2} = \frac{2x(x-1) – (x^2+1)(1)}{(x-1)^2} = \frac{2x^2 – 2x – x^2 – 1}{(x-1)^2} = \frac{x^2 – 2x – 1}{(x-1)^2}\)
\(\frac{dy}{dx} = \frac{u’v – uv’}{v^2} = \frac{2x(x-1) – (x^2+1)(1)}{(x-1)^2} = \frac{2x^2 – 2x – x^2 – 1}{(x-1)^2} = \frac{x^2 – 2x – 1}{(x-1)^2}\)
✅ \(\frac{dy}{dx} = \frac{x^2 – 2x – 1}{(x-1)^2}\)
Q4 Differentiate \(y = \sin(3x^2)\) using chain rule.
Let \(u = 3x^2\), \(\frac{du}{dx} = 6x\)
\(\frac{dy}{dx} = \cos u \cdot \frac{du}{dx} = \cos(3x^2) \cdot 6x = 6x \cos(3x^2)\)
\(\frac{dy}{dx} = \cos u \cdot \frac{du}{dx} = \cos(3x^2) \cdot 6x = 6x \cos(3x^2)\)
✅ \(\frac{dy}{dx} = 6x \cos(3x^2)\)
Q5 Differentiate \(y = e^{2x} \sin x\) using product rule.
Let \(u = e^{2x}\), \(v = \sin x\), \(u’=2e^{2x}\), \(v’=\cos x\)
\(\frac{dy}{dx} = u’v + uv’ = 2e^{2x}\sin x + e^{2x}\cos x = e^{2x}(2\sin x + \cos x)\)
\(\frac{dy}{dx} = u’v + uv’ = 2e^{2x}\sin x + e^{2x}\cos x = e^{2x}(2\sin x + \cos x)\)
✅ \(\frac{dy}{dx} = e^{2x}(2\sin x + \cos x)\)
Q6 Differentiate \(y = \ln(x^2 + 3x + 1)\).
\(\frac{dy}{dx} = \frac{1}{x^2+3x+1} \cdot (2x+3) = \frac{2x+3}{x^2+3x+1}\)
✅ \(\frac{dy}{dx} = \frac{2x+3}{x^2+3x+1}\)
Q7 Differentiate \(y = \sinh(2x)\).
\(\frac{dy}{dx} = \cosh(2x) \cdot 2 = 2\cosh(2x)\)
✅ \(\frac{dy}{dx} = 2\cosh(2x)\)
Q8 Differentiate \(y = \cosh^2 x\).
\(\frac{dy}{dx} = 2\cosh x \cdot \sinh x = \sinh(2x)\)
✅ \(\frac{dy}{dx} = \sinh(2x)\)
Q9 Find the second derivative of \(y = x^4 – 3x^2 + 2x\).
\(y’ = 4x^3 – 6x + 2\)
\(y” = 12x^2 – 6\)
\(y” = 12x^2 – 6\)
✅ \(y” = 12x^2 – 6\)
Q10 Differentiate \(y = x^x\) using logarithmic differentiation.
\(\ln y = x \ln x\)
\(\frac{1}{y}\frac{dy}{dx} = \ln x + x \cdot \frac{1}{x} = \ln x + 1\)
\(\frac{dy}{dx} = y(\ln x + 1) = x^x(\ln x + 1)\)
\(\frac{1}{y}\frac{dy}{dx} = \ln x + x \cdot \frac{1}{x} = \ln x + 1\)
\(\frac{dy}{dx} = y(\ln x + 1) = x^x(\ln x + 1)\)
✅ \(\frac{dy}{dx} = x^x(\ln x + 1)\)
Q11 Differentiate \(y = \tan^{-1}(x^2)\).
\(\frac{dy}{dx} = \frac{1}{1+(x^2)^2} \cdot 2x = \frac{2x}{1+x^4}\)
✅ \(\frac{dy}{dx} = \frac{2x}{1+x^4}\)
Q12 Differentiate \(y = \frac{\ln x}{x}\).
\(\frac{dy}{dx} = \frac{(1/x)\cdot x – \ln x \cdot 1}{x^2} = \frac{1 – \ln x}{x^2}\)
✅ \(\frac{dy}{dx} = \frac{1 – \ln x}{x^2}\)
Q13 Differentiate \(y = (x^2+1)^5\) using chain rule.
\(\frac{dy}{dx} = 5(x^2+1)^4 \cdot 2x = 10x(x^2+1)^4\)
✅ \(\frac{dy}{dx} = 10x(x^2+1)^4\)
Q14 Differentiate \(y = e^{\sin x}\).
\(\frac{dy}{dx} = e^{\sin x} \cdot \cos x = \cos x \cdot e^{\sin x}\)
✅ \(\frac{dy}{dx} = e^{\sin x} \cos x\)
Q15 Differentiate \(y = \sec(2x)\) and find \(y”\).
\(y’ = \sec(2x)\tan(2x) \cdot 2 = 2\sec(2x)\tan(2x)\)
\(y” = 2[2\sec(2x)\tan(2x)\tan(2x) + \sec(2x)\sec^2(2x)\cdot 2] = 4\sec(2x)\tan^2(2x) + 4\sec^3(2x)\)
\(y” = 2[2\sec(2x)\tan(2x)\tan(2x) + \sec(2x)\sec^2(2x)\cdot 2] = 4\sec(2x)\tan^2(2x) + 4\sec^3(2x)\)
✅ \(y’ = 2\sec(2x)\tan(2x)\)
✏️ Practice Questions
P1 Differentiate \(y = 4x^3 – 5x^2 + 2x – 9\)
✅ Answer: \(12x^2 – 10x + 2\)
P2 Differentiate \(y = (3x+2)(x^2-5)\) using product rule
✅ Answer: \(9x^2 + 4x – 15\)
P3 Differentiate \(y = \frac{2x+1}{x-3}\) using quotient rule
✅ Answer: \(\frac{-7}{(x-3)^2}\)
P4 Differentiate \(y = \cos(4x^3)\) using chain rule
✅ Answer: \(-12x^2 \sin(4x^3)\)
P5 Differentiate \(y = \tanh(3x)\)
✅ Answer: \(3 \text{sech}^2(3x)\)
📋 Multiple Choice Questions (HSEB Pattern)
MCQ 1 The derivative of \(x^n\) with respect to x is:
A) \(nx^{n-1}\) B) \(nx^{n+1}\) C) \(nx^n\) D) \(x^{n-1}\)
✅ Answer: A) \(nx^{n-1}\)
MCQ 2 The derivative of \(\sin x\) is:
A) \(\cos x\) B) \(-\cos x\) C) \(\sec^2 x\) D) \(-\sin x\)
✅ Answer: A) \(\cos x\)
MCQ 3 The derivative of \(\cos x\) is:
A) \(\sin x\) B) \(-\sin x\) C) \(\tan x\) D) \(-\cos x\)
✅ Answer: B) \(-\sin x\)
MCQ 4 The product rule states \(\frac{d}{dx}(uv) =\)
A) \(u’v’\) B) \(u’v + uv’\) C) \(u’v – uv’\) D) \(\frac{u’v – uv’}{v^2}\)
✅ Answer: B) \(u’v + uv’\)
MCQ 5 The derivative of \(\ln x\) is:
A) \(x\) B) \(\frac{1}{x}\) C) \(\frac{1}{x^2}\) D) \(e^x\)
✅ Answer: B) \(\frac{1}{x}\)
MCQ 6 The derivative of \(e^x\) is:
A) \(e^x\) B) \(xe^{x-1}\) C) \(\ln x\) D) \(x e^x\)
✅ Answer: A) \(e^x\)
MCQ 7 The derivative of \(\sinh x\) is:
A) \(\sinh x\) B) \(\cosh x\) C) \(-\cosh x\) D) \(\tanh x\)
✅ Answer: B) \(\cosh x\)
MCQ 8 The second derivative of \(x^3\) is:
A) \(3x^2\) B) \(6x\) C) \(6\) D) \(0\)
✅ Answer: B) \(6x\)
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This PDF provides the solutions of every question from the 1st exercise of class 12 derivatives chapter. Since this chapter has only one exercise, you don’t have to search for other exercises.
This PDF provides the solutions of every question from the 1st exercise of class 12 derivatives chapter. Since this chapter has only one exercise, you don’t have to search for other exercises.
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📌 Key Formulas Summary:
• Power Rule: \(\frac{d}{dx}(x^n) = nx^{n-1}\)
• Product Rule: \(\frac{d}{dx}(uv) = u’v + uv’\)
• Quotient Rule: \(\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u’v – uv’}{v^2}\)
• Chain Rule: \(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\)
• Trigonometric Functions: \(\frac{d}{dx}\sin x = \cos x\), \(\frac{d}{dx}\cos x = -\sin x\)
• Exponential/Log: \(\frac{d}{dx}e^x = e^x\), \(\frac{d}{dx}\ln x = \frac{1}{x}\)
• Hyperbolic: \(\frac{d}{dx}\sinh x = \cosh x\), \(\frac{d}{dx}\cosh x = \sinh x\)
• Power Rule: \(\frac{d}{dx}(x^n) = nx^{n-1}\)
• Product Rule: \(\frac{d}{dx}(uv) = u’v + uv’\)
• Quotient Rule: \(\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u’v – uv’}{v^2}\)
• Chain Rule: \(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\)
• Trigonometric Functions: \(\frac{d}{dx}\sin x = \cos x\), \(\frac{d}{dx}\cos x = -\sin x\)
• Exponential/Log: \(\frac{d}{dx}e^x = e^x\), \(\frac{d}{dx}\ln x = \frac{1}{x}\)
• Hyperbolic: \(\frac{d}{dx}\sinh x = \cosh x\), \(\frac{d}{dx}\cosh x = \sinh x\)