Differentiation Questions

Introduction

Differentiation is an important concept in calculus that helps us understand how a function changes. It is used to find the rate of change of one quantity with respect to another. For example, if we want to know how quickly the distance is changing with respect to time, we use differentiation.

The process of differentiation gives us a derivative, which represents the slope of a curve at a given point. In simple words, it tells us how steep a graph is or how quickly something increases or decreases. Differentiation is widely used in many real-life situations, such as mathematics, physics, economics, speed, development, and motion.

In this topic, we will explore the basic rules of differentiation formulas, common formulas, solved examples, and different types of questions that are often shown in the exam. This step-by-step guide will make it easy for you to understand and practise different questions effectively.

Table of Contents

• Definition
• Formulas in Differentiation
• Questions on Differentiation
• Solved Examples
• Practice Questions
• FAQs on Differentiation Questions
  
  

Definition

Differentiation is the process of finding the rate of change of a function. In simple words, it tells us how fast one quantity changes when another quantity changes.

For example:

• If distance changes with time, differentiation helps us find speed.

• If speed changes with time, differentiation helps us find acceleration.

 

Formulas in Differentiation

• $\frac{d}{dx}\left( x^{n} \right)= n . x^{n-1}$

• $\frac{d}{dx}\left( sin x \right)= cos x$

• $\frac{d}{dx}\left(cos x \right)= - sinx$

• $\frac{d}{dx}\left(tanx \right)= sec^{2}x$

• $\frac{d}{dx}\left(In x \right)= \frac{1}{x}$

• $\frac{d}{dx}\left(e^{x} \right)= e^{x}$

• Product Rule: $\frac{d}{dx}\left( u .v \right)= u'v + uv'$

• Quotient Rule: $\frac{d}{dx}\left( \frac{u}{v}\right)= \frac{u'v - uv'}{v^{2}}$

• Chain Rule: $\frac{d}{dx}\left( f\left( g\left( x \right) \right)\right)= f'\left( g\left( x \right) \right). g'\left( x \right)$

 

Questions on Differentiation

1. 1. Differentiate: $f\left( x \right)= 7x^{3}-5x+6$

          Apply the power rule: $\frac{d }{dx}\left( x^{n} \right)= n \left( x^{n-1} \right)$

          Differentiate $7x^{3}$: $7 \left( 3x^{2} \right)= 21x^{2}$

          Differentiate $-5x$: $-5$

          Constant 6 becomes 0

          Final Answer: $f'\left( x \right)= 21x^{2}-5$

2.  

3.  

   2. Find $\frac{dy }{dx}$: $y = cos(x)$

          Use the standard derivative: $\frac{d }{dx}\left( cos x \right)= - sin x$

          Final Answer: $\frac{d y}{dx} = - sin x$

4.  

5.  

   3. Differentiate: $y = In \left( 3x + 1 \right)$

          Use chain rule: $\frac{d }{dx}\left[ In\left( u \right) \right]= \frac{1}{u}\left( \frac{du }{dx} \right)$

          Let $u = 3x + 1$, then $\frac{d u}{dx}= 3$

          Apply the rule: $\frac{1}{3x + 1}. 3$

          Final Answer: $\frac{d y}{dx}= \frac{3}{3x+1}$

6.  

7.  

   4. Find the derivative: $y = x^{2}.e^{x}$

          Use the product rule: $\frac{d }{dx}\left( u.v \right)= u'v+uv'$

          Let $u = x^{2},v = e^{x}$

          $u' = 2x , v' = e^{x}$

          Apply rule: $2x. e^{x}+ x^{2}.e^{x}$

          Factor if needed: $\frac{d y}{dx}= e^{x}\left( 2x + x^{2} \right)$

8.  

9.  

   5. Differentiate: $y = \frac{x^{2}+1}{x-2}$

          Use quotient rule: $\frac{d }{dx}\left[ \frac{u}{v} \right]= \frac{\left( u'v-uv' \right)}{v^{2}}$

          Let $u = x^{2}+1 , v = x -2$

          $u' = 2x, v' = 1$

          Apply: $\frac{\left[ \left( 2x \right)\left( x-2 \right) - \left( x^{2} +1\right).1\right]}{\left( x-2 \right)^{2}}$

          Simplify numerator: $2x^{2}-4x-x^{2}-1 = x^{2} -4x -1$

          Final Answer: $\frac{d y}{dx}= \frac{x^{2}-4x -1}{\left( x-2 \right)^{2}}$

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11.  

    6. Find $\frac{d y}{dx}$: $y=\left( sin\left( x \right) \right)^{2}$

           Use chain rule: $\frac{d }{dx}\left[ \left( sin x \right) ^{2}\right]= 2. sin\left( x \right).\frac{d }{dx}\left[ sin\left( x \right) \right]$

           $\frac{d }{dx}\left( sin \left( x\right) \right) = cos \left( x \right)$

           Final Answer: $\frac{dy }{dx}= 2. sin\left( x \right).cos\left( x \right)$

12.  

13.  

    7. Differentiate: $y = \frac{5}{x^{2}}$

           Rewrite as $y = {5}{x^{-2}}$

           Use power rule: $\frac{d }{dx}\left( x^{n} \right)= n . x^{n-1}$

           $\frac{d y}{dx}= 5 . \left( -2 \right)x^{-3}= -\frac{10}{x^{3}}$

           Final Answer: $\frac{d y}{dx}= -\frac{10}{x^{3}}$

14.  

15.  

    8. Find the derivative: $y = arctan\left( x \right)$

           Use the standard formula: $\frac{d }{dx}arctan\left( x \right)= \frac{1}{1+x^{2}}$

           Final Answer: $\frac{d y}{dx}= \frac{1}{1+x^{2}}$

16.  

17.  

    9. Differentiate: $y= e^{2x}$

           Use chain rule: $\frac{d }{dx} e^{2x}=e^{2x} . \frac{d }{dx}\left( 2x \right)$

           $\frac{d }{dx}\left( 2x \right)=2$

           Final Answer: $\frac{dy }{dx} =2e^{2x}$

18.  

19.  

    10. Find $\frac{d y}{dx}:y = \left( x^{2} +4x+5\right)^{4}$

           Use chain rule: $\frac{d }{dx}\left( u^{4} \right)=4u^{3}. \frac{d u}{dx}$

           Let $u= x^{2}+4x+5$, then $\frac{d u}{dx}=2x+4$

           $\frac{d y}{dx}=4\left( x^{2} +4x+5\right)^{3}. \left( 2x+4 \right)$

           Final Answer: $\frac{d y}{dx}=4\left( 2x +4\right)\left( x^{2} +4x+5\right)^{3}$
    
    

 

Solved Examples

1. Differentiate: $f\left( x \right)= x^{5}$

• Use the Power Rule: $\frac{d }{dx}\left( x^{n} \right)= n\cdot x^{n-1}$

• $f'\left( x \right)= 5\cdot x^{4}$

Answer: $f'\left( x \right)= 5\cdot x^{4}$

 

2. Differentiate: $f\left( x \right)= 3x^{2}+2x+1$

• $\frac{d }{dx}\left( 3x^{2} \right)=6x$

• $\frac{d }{dx}\left( 2x \right)=2$

• $\frac{d }{dx}\left( 1\right)=0$

Answer: $f'\left( x \right)= 6x +2$

 

3. Differentiate: $y = sin \left( x \right)$

• $\frac{d }{dx}sin\left( x \right)= cos \left( x \right)$

Answer: $\frac{dy }{dx}= cos \left( x \right)$

 

4. Differentiate: $y = In \left( x \right)$

• $\frac{d }{dx}\left( In\left( x \right) \right)= \frac{1}{x}$

Answer: $\frac{d y}{dx}= \frac{1}{x}$

 

5. Differentiate: $y = e^{x}$

• $\frac{d }{dx}e^{x} = e^{x}$

Answer: $\frac{dy }{dx}= e^{x}$

 

6. Differentiate: $y = x^{3 }\cdot sin \left( x \right)$

• Use the product rule: $\left( uv \right)' = u'v + uv'$

• $u = x^{3}\to u'= 3x^{2}$

• $v = sin \left( x \right)\to v' = cos \left( x \right)$

• $\frac{d y}{dx} = 3x^{2}\cdot sin \left( x \right)+ x^{3}\cdot cos \left( x \right)$

Answer: $\frac{d y}{dx} = 3x^{2}\cdot sin \left( x \right)+ x^{3}\cdot cos \left( x \right)$

 

7. Differentiate: $y = \frac{2x^{2}+ 1 }{x+3}$

• Use Quotient Rule: $\left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^{2}}$

• $u = 2x^{2}+ 1 \to u ' = 4x$

• $v = x + 3 \to v' = 1$

• $\frac{dy}{dx} = \frac{(4x)(x + 3) - (2x^{2} + 1)(1)}{(x + 3)^{2}}$

Answer: $\frac{dy}{dx} = \frac{(4x)(x + 3) - (2x^{2} + 1)}{(x + 3)^{2}}$

 

8. Differentiate: $y =\left( 3x + 4\right) ^{2}$

• Use the chain rule: $\frac{d}{dx}\left[u^{2}\right] = 2u \cdot \frac{du}{dx}$

• $u = 3x + 4 \;\;\;\Rightarrow\;\;\; \frac{du}{dx} = 3$

• $\frac{dy}{dx} = 2(3x + 4)\cdot 3 = 6(3x + 4)$

Answer: $\frac{dy}{dx} = 6(3x + 4)$

 

9. Differentiate: $y = \sqrt{x}$

• Rewrite: $y = x^{\frac{1}{2}}$

• Use the power rule: $\frac{d}{dx}\left(x^n\right) = n \cdot x^{n-1}$

• $\frac{dy}{dx} = \frac{1}{2}x^{-\tfrac{1}{2}} = \frac{1}{2\sqrt{x}}$

Answer: $\frac{dy}{dx} = \frac{1}{2\sqrt{x}}$

 

10. Differentiate: $y = tan(x)$

• $\frac{d}{dx} \left[ \tan(x) \right] = \sec^2(x)$

Answer: $\frac{dy}{dx} = \sec^2(x)$

Practice Questions

1. Differentiate: $f(x) = 7x^3 - 5x + 6$

2. Find $\frac{d y}{dx} : y = cos \left( x \right)$

3. Differentiate: $y = \ln(3x + 1)$

4. Find the derivative: $y = x^2 \cdot e^x$

5. Differentiate: $y = \frac{x^2 + 1}{x - 2}$

6. Find $\frac{d y}{dx} : y = \left(sin \left( x \right) \right)^{2}$

7. Differentiate: $y = \frac{5}{x^2}$

8. Find the derivative: $y = \arctan(x)$

9. Differentiate: $y = e^{2x}$

10. Find $\frac{d y}{dx} : y = \left( x^{2}+4x+5 \right)^{4}$

 

Frequently Asked Questions on Differentiation Questions

1. What is the purpose of differentiation in mathematics?

Answer: A function's rate of change can be measured via differentiation. When we calculate acceleration (rate of change of velocity) or velocity (rate of change of position), for example, it helps us comprehend how one quantity changes in relation to another. In order to analyse dynamic systems, differentiation is a fundamental idea in physics, economics, and engineering.

 

2. How is differentiation applied in real life?

Answer: There are several practical uses for differentiation. It aids in the computation of acceleration, velocity, and speed in physics. It can be applied in economics to either maximise profit or minimise expenses. Differentiation aids in the modelling of population growth rates in biology. It is also frequently used in engineering to examine structural stability and forces.

 

3. How do I differentiate between simple and complex functions?

Answer: Simple methods like the power rule can be used to differentiate simple functions, such as polynomials (x², for example). Logarithmic, exponential, and trigonometric functions are examples of complex functions that may involve a combination of rules, such as the quotient, product, or chain rules. Divide complicated functions into smaller components, then apply the relevant rule to each one.

 

4. What is the significance of the derivative in real-world problems?

Answer: The derivative provides information on a function's change over time, which is essential for resolving practical issues. In the corporate world, for example, it can assist in figuring out the best prices for goods. It indicates the speed of an object at any given moment in physics. It can be used to compute material stress and strain in engineering.