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# Lecture - 9 Chapter 13 Limits and Derivatives

All rules for direct differentiation Detailed Explanation and Tricks for direct derivative in case of, Addition and Subtraction, Multiplication (Product Rule)

NCERT EXERCISE 13.2 |

**Question 11. ** Find the derivative of the following functions:**(i).** \(\sin x \cos x\)**(ii).** \(\sec x\)**(iii).** \(5 \sec x + 4 \cos x\)**(iv).** \(\cosec x\)**(v).** \(3 \cot x + 5 \cosec x\)**(vi).** \(5 \sin x – 6 \cos x + 7\)**(vii).** \(2 \tan x – 7 \sec x\)

**Question 9. **Find the derivative of**(i).** \(2x – \frac{3}{4}\)**(ii).** \((5x^3+3x-1)(x-1)\)**(iii).** \(x^{-3}(5+3x)\)**(iv).** \(x^5 (3-6x^{-9})\)**(v).** \(x^{-4}(3-4x^{-5})\)**(vi).** \(\frac{2}{x+1}-\frac{x^2}{3x-1}\)

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Division (Quotient Rule)

**Question 8. **Find the derivative of \( \frac{x^n-a^n}{x-1}\) for some constant *a.*

**Question 7. ** For some constants *a* and *b*, find the derivative of**(i).** \((x-a)(x-b)\)**(ii).** \((ax^2+b)^2\)**(iii).** \(\frac{x-a}{x-b}\)

**Question 6. **Find the derivative of \( x^n+ax^{n-1}+a^2x^{n-2}+ \dots + a^{n-1}x+a^n\) for some fixed real number *a*.

**Question 5. ** For the function \( f(x) = \frac{x^{100}}{100}+\frac{x^{99}}{99}+ \dots + \frac{x^2}{2}+x+1\)

Prove that \(f'(1)=100f'(0)\).

**Question 1.** Find the derivative of \(x^2-2\) at \( x= 10\).

**Question 2.** Find the derivative of \(x\) at \( x= 1\).

**Question 3.** Find the derivative of \(99x\) at \( x= 100\).