03_Ex 2.3

Exercise Questions

Question.	Question Links
Q1. Differentiate $x^{4}+2 x^{3}+x^{2}$	2.3 Q-1
Q2. Differentiate $x^{-3}+2 x^{\frac{3}{2}}+3$	2.3 Q-2
Q3. Differentiate $\frac{a+x}{a-x}$	2.3 Q-3
Q4. Differentiate $\frac{2 x-3}{2 x+1}$	2.3 Q-4
Q5. Differentiate $(x-5)(3-x)$	2.3 Q-5
Q6. Differentiate $\left(\sqrt{x}-\frac{1}{\sqrt{x}}\right)^{2}$	2.3 Q-6
Q7. Differentiate $\frac{(1+\sqrt{x})\left(x-x^{\frac{5}{2}}\right)}{\sqrt{x}}$	2.3 Q-7
Q8. Differentiate $\frac{\left(x^{2}+1\right)^{2}}{x^{2}-1}$	2.3 Q-8
Q9. Differentiate $\frac{x^{2}+1}{x^{2}-3}$	2.3 Q-9
Q10. Differentiate $\frac{\sqrt{1+x}}{\sqrt{1-x}}$	2.3 Q-10
Q11. Differentiate $\frac{2 x-1}{\sqrt{x^{2}}+1}$	2.3 Q-11
Q12. Differentiate $\sqrt{\frac{a-x}{a+-x}}$	2.3 Q-12
Q13. Differentiate $\frac{\sqrt{x^{2}+1}}{\sqrt{x^{2}-1}}$	2.3 Q-13
Q14. Differentiate $\frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}}$	2.3 Q-14
Q15. Differentiate $\frac{x \sqrt{a+x}}{\sqrt{a-x}}$	2.3 Q-15
Q16. If $y=\sqrt{x}-\frac{1}{\sqrt{x}}$, show that $2 x \frac{d y}{d x}+y=2 \sqrt{x}$	2.3 Q-16
Q17. If $y=x^{4}+2 x^{2}+2$, Prove that $\frac{d y}{d x}=4 x \sqrt{y-1}$	2.3 Q-17

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Overview

This exercise focuses on applying basic differentiation rules to solve problems. These rules include the Power Rule, Product Rule, Quotient Rule, and other standard differentiation techniques that allow for the computation of derivatives in various forms. Understanding these methods is crucial for solving a wide range of mathematical and real-world problems involving rates of change.

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Key Concepts

1. Constant Multiple Rule:
   $\frac {d}{dx} [c \cdot f(x)] = c \cdot \frac {d}{dx} f(x)$
   The constant factor can be taken outside the derivative.

2. Sum or Difference Rule:
   $\frac {d}{dx} [f(x) \pm g(x)] = \frac {d}{dx} f(x) \pm \frac {d}{dx} g(x)$
   Derivatives of sums or differences can be computed separately.

3. Power Rule:
   $\frac {d}{dx} [f(x)^n] = n \cdot f(x)^{n-1} \cdot \frac {d}{dx} f(x)$
   When differentiating a function raised to a power, first multiply by the power, then reduce the exponent by 1.

4. Product Rule:
   $\frac {d}{dx} [f(x) \cdot g(x)] = f(x) \cdot \frac {d}{dx} g(x) + g(x) \cdot \frac {d}{dx} f(x)$
   The derivative of a product involves the derivative of each function multiplied by the other function.

5. Quotient Rule:
   $\frac{d}{dx} \frac{f(x)}{g(x)} = \frac{g(x) \cdot \frac{d}{dx} f(x) - f(x) \cdot \frac{d}{dx} g(x)}{[g(x)]^2}$
   To differentiate a quotient, use this rule to find the derivative of the numerator and denominator.

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Important Formulas

• Constant Multiple Rule:
  $\frac {d}{dx} c \cdot f(x) = c \cdot \frac {d}{dx} f(x)$

• Sum or Difference Rule:
  $\frac {d}{dx} [f(x) \pm g(x)] = \frac {d}{dx} f(x) \pm \frac {d}{dx} g(x)$

• Power Rule:
  $\frac {d}{dx} [f(x)^n] = n \cdot f(x)^{n-1} \cdot \frac {d}{dx} f(x)$

• Product Rule:
  $\frac {d}{dx} [f(x) \cdot g(x)] = f(x) \cdot \frac {d}{dx} g(x) + g(x) \cdot \frac {d}{dx} f(x)$

• Quotient Rule:
  $\frac{d}{dx} \frac{f(x)}{g(x)} = \frac{g(x) \cdot \frac{d}{dx} f(x) - f(x) \cdot \frac{d}{dx} g(x)}{[g(x)]^2}$

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Tips and Tricks

• For the Power Rule, remember that when differentiating any function raised to a power, the exponent becomes a multiplier, and the exponent is reduced by one.

• The Product Rule and Quotient Rule are crucial for handling multiplication and division of functions. Always apply them carefully to avoid mistakes, especially when dealing with complex expressions.

• When using the Quotient Rule, carefully differentiate both the numerator and denominator and remember the negative sign in the numerator.

• Simplify the expression as much as possible after differentiating to make the solution clearer and easier to interpret.

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Summary

This exercise focuses on applying various differentiation rules such as the Constant Multiple Rule, Sum/Difference Rule, Power Rule, Product Rule, and Quotient Rule to solve derivatives. Understanding these rules and applying them correctly is essential for solving problems involving rates of change, particularly in calculus. Practice with these rules will help develop fluency in differentiation techniques.

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Reference

By Sir Shahzad Sair:

By Great Science Academy: