Higher Order Derivatives

1. Statement of the Concept

If a function [y = f(x)] is differentiable, then:

• The first derivative is:
  [\dfrac{d}{dx}(y) = \dfrac{dy}{dx}]
• The second derivative is:
  [\dfrac{d}{dx}(\dfrac{dy}{dx}) = \dfrac{d^2y}{dx^2}]
• The third derivative is:
  [\dfrac{d}{dx}(\dfrac{d^2y}{dx^2}) = \dfrac{d^3y}{dx^3}]

And similarly, the nth derivative is:

[\dfrac{d^n y}{dx^n}]

These derivatives are called Higher Order Derivatives.

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2. Explanation and Mathematical Derivation

The first derivative represents the rate of change of the function.

The second derivative gives the rate of change of the first derivative, indicating curvature or concavity.

Differentiating multiple times leads to higher order derivatives.

General formula:

If [y = f(x)]

[\dfrac{d^n y}{dx^n}][ = \dfrac{d}{dx}(\dfrac{d^{n-1}y}{dx^{n-1}})]

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3. Examples with Step by Step Solutions

Example 1 — Polynomial (4th derivative)

Find [\dfrac{d^{4}y}{dx^{4}}] for [y = x^{6}].

Step-by-step solution:

1. [\dfrac{d}{dx}(x^{6}) = 6x^{5}]
2. [\dfrac{d^{2}}{dx^{2}}(x^{6}) ][= \dfrac{d}{dx}(6x^{5}) = 30x^{4}]
3. [\dfrac{d^{3}}{dx^{3}}(x^{6}) ][= \dfrac{d}{dx}(30x^{4}) = 120x^{3}]
4. [\dfrac{d^{4}}{dx^{4}}(x^{6}) = \dfrac{d}{dx}(120x^{3}) = 360x^{2}]

Conclusion:
[\dfrac{d^{4}y}{dx^{4}} = 360x^{2}].

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Example 2 — Exponential (general n-th derivative)

Find [\dfrac{d^{n}y}{dx^{n}}] for [y = e^{3x}], give first few derivatives and general formula.

Step-by-step solution:

1. [\dfrac{d}{dx}(e^{3x}) = 3e^{3x}]
2. [\dfrac{d^{2}}{dx^{2}}(e^{3x})][ = \dfrac{d}{dx}(3e^{3x}) = 3^{2}e^{3x}]
3. [\dfrac{d^{3}}{dx^{3}}(e^{3x})][ = 3^{3}e^{3x}]
4. By induction, the general pattern is:
   [\dfrac{d^{n}}{dx^{n}}(e^{3x})][ = 3^{n}e^{3x}]

Conclusion:
[\dfrac{d^{n}y}{dx^{n}} = 3^{n}e^{3x}].

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Example 3 — Trigonometric (3rd derivative)

Find [\dfrac{d^{3}y}{dx^{3}}] for [y = \sin(2x)].

Step-by-step solution:

1. [\dfrac{d}{dx}(\sin(2x)) = 2\cos(2x)] (chain rule)
2. [\dfrac{d^{2}}{dx^{2}}(\sin(2x)) ][= \dfrac{d}{dx}(2\cos(2x)) ][= 2\cdot(-2\sin(2x)) = -4\sin(2x)]
3. [\dfrac{d^{3}}{dx^{3}}(\sin(2x)) ][= \dfrac{d}{dx}(-4\sin(2x)) ][= -4\cdot 2\cos(2x) = -8\cos(2x)]

Conclusion:
[\dfrac{d^{3}y}{dx^{3}} = -8\cos(2x)].

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Example 4 — Product (second derivative)

Find [\dfrac{d^{2}y}{dx^{2}}] for [y = x^{2}e^{x}].

Step-by-step solution:

1. First derivative using product rule:
   [\dfrac{d}{dx}(x^{2}e^{x}) ][= x^{2}\dfrac{d}{dx}(e^{x}) + e^{x}\dfrac{d}{dx}(x^{2}) ][= x^{2}e^{x} + e^{x}\cdot 2x ][= e^{x}(x^{2} + 2x)]
2. Second derivative — differentiate [e^{x}(x^{2}+2x)] using product rule again:
   Let [u=e^{x}], [v=(x^{2}+2x)] so [u’=e^{x}], [v’=2x+2].
   Then [\dfrac{d^{2}y}{dx^{2}} = u’v + uv’ ][= e^{x}(x^{2}+2x) + e^{x}(2x+2)]
3. Factor [e^{x}]:
   [\dfrac{d^{2}y}{dx^{2}} ][= e^{x}\big(x^{2}+2x + 2x + 2\big) ][= e^{x}\big(x^{2} + 4x + 2\big)]

Conclusion:
[\dfrac{d^{2}y}{dx^{2}} ][= e^{x}\big(x^{2} + 4x + 2\big)].

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Example 5 — Logarithmic (4th derivative)

Find [\dfrac{d^{4}y}{dx^{4}}] for [y = \ln x].

Step-by-step solution:

1. [\dfrac{d}{dx}(\ln x) = \dfrac{1}{x}]
2. [\dfrac{d^{2}}{dx^{2}}(\ln x) ][= \dfrac{d}{dx}\big(x^{-1}\big) ][= -x^{-2} ][= -\dfrac{1}{x^{2}}]
3. [\dfrac{d^{3}}{dx^{3}}(\ln x) ][= \dfrac{d}{dx}\big(-x^{-2}\big) ][= 2x^{-3} ][= \dfrac{2}{x^{3}}]
4. [\dfrac{d^{4}}{dx^{4}}(\ln x) ][= \dfrac{d}{dx}\big(2x^{-3}\big) ][= 2\cdot(-3)x^{-4} ][= -6x^{-4} ][= -\dfrac{6}{x^{4}}]

Conclusion:
[\dfrac{d^{4}y}{dx^{4}} ][= -\dfrac{6}{x^{4}}], valid for [x\ne 0].

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4. Key Features

1. Higher derivatives give detailed behavior of a function.
2. Second derivative tells concavity and points of inflection.
3. Third derivative helps with advanced curve analysis.
4. Used in physics: velocity → acceleration → jerk → jounce…
5. nth derivatives appear in differential equations, Taylor series.
6. Some functions become zero after finite derivatives.

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5. Important Formulas to Remember

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6. Conceptual Questions with Solutions

1. What does the second derivative represent?

It represents concavity and acceleration of change.

2. Why do some functions have zero higher derivatives?

Because polynomial degrees reduce with each differentiation.

3. What does nth derivative mean?

It is the result of differentiating a function n times.

4. Why are higher derivatives useful in physics?

They help study motion: velocity, acceleration, jerk, etc.

5. Does nth derivative always exist?

No, only if the function remains differentiable enough times.

6. Is the nth derivative unique?

Yes, differentiation is deterministic for differentiable functions.

7. Does nth derivative apply to trigonometric functions?

Yes, they show periodic derivative patterns.

8. How do higher derivatives relate to Taylor Series?

Coefficients of the Taylor series are derived from higher derivatives.

9. Do derivatives always reduce function complexity?

Not always — exponential functions remain exponential.

10. Why is nth derivative of [e^{ax}] easy?

Because differentiating preserves its form; only coefficient multiplies.

11. Does [\dfrac{d^n y}{dx^n}] affect domain?

Yes, differentiability must hold over the domain.

12. Can nth derivative be expressed in general formulas?

Yes, e.g., power rule and formulas for trig/exponential functions.

13. Are higher derivatives used in engineering?

Yes, especially for motion dynamics and system modeling.

14. What happens to derivative of a constant?

First derivative is zero, and all subsequent derivatives remain zero.

15. Why is differentiation called order-based?

Because each successive derivative is indexed by an order number.

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7. FAQ / Common Misconceptions

1. Higher derivatives always become zero.

This is true only for polynomials.

2. The second derivative always exists if the first exists.

Not true — differentiability must hold again.

3. nth derivative notation always indicates power n.

No, it denotes the order of differentiation.

4. Higher derivatives have no real meaning.

They represent dynamic behavior of functions in science.

5. nth derivative cannot be negative.

It can — signs depend on the function type.

6. nth derivative is always simpler.

Some become more complex (e.g., products of functions).

7. Second derivative alone tells the entire shape.

Higher orders help understand more complex curvature behavior.

8. Higher derivatives are not needed in real life.

Used everywhere: robotics, aerospace engineering, economics, etc.

9. Only polynomials have nth derivatives.

Trig, exponential, hyperbolic functions also differentiate infinitely.

10. Higher derivatives don’t appear in equations.

They form the basis of differential equations and series.

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8. Practice Questions (Step-by-Step Solutions)

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Question 1

Find [\dfrac{d^3y}{dx^3}] if [y = x^5].

Step-by-Step Solution:

1. [\dfrac{dy}{dx} = 5x^4]
2. [\dfrac{d^2y}{dx^2} = 20x^3]
3. [\dfrac{d^3y}{dx^3} = 60x^2]

Conclusion: [\dfrac{d^3y}{dx^3} = 60x^2]

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Question 2

Find the 4th derivative of [y = e^{3x}].

Steps:

1. [\dfrac{dy}{dx}=3e^{3x}]
2. [\dfrac{d^2y}{dx^2}=9e^{3x}]
3. [\dfrac{d^3y}{dx^3}=27e^{3x}]
4. [\dfrac{d^4y}{dx^4}=81e^{3x}]

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Question 3

Find [\dfrac{d^3y}{dx^3}] if [y = \sin(2x)].

1. [\dfrac{dy}{dx}=2\cos(2x)]
2. [\dfrac{d^2y}{dx^2}=-4\sin(2x)]
3. [\dfrac{d^3y}{dx^3}=-8\cos(2x)]

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Question 4

Find 3rd derivative of [y = x^4 + x^2].

1. [\dfrac{dy}{dx} = 4x^3 + 2x]
2. [\dfrac{d^2y}{dx^2} = 12x^2 + 2]
3. [\dfrac{d^3y}{dx^3} = 24x]

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Question 5

Find [\dfrac{d^2y}{dx^2}] if [y = \cos x].

1. [\dfrac{dy}{dx} = -\sin x]
2. [\dfrac{d^2y}{dx^2} = -\cos x]

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Question 6

Find [\dfrac{d^4y}{dx^4}] if [y = \ln x].

1. [\dfrac{dy}{dx} = \dfrac{1}{x}]
2. [\dfrac{d^2y}{dx^2} = -\dfrac{1}{x^2}]
3. [\dfrac{d^3y}{dx^3} = \dfrac{2}{x^3}]
4. [\dfrac{d^4y}{dx^4} = -\dfrac{6}{x^4}]

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Question 7

Find 5th derivative of [y = x^7].

1. [\dfrac{dy}{dx} = 7x^6]
2. [\dfrac{d^2y}{dx^2} = 42x^5]
3. [\dfrac{d^3y}{dx^3} = 210x^4]
4. [\dfrac{d^4y}{dx^4} = 840x^3]
5. [\dfrac{d^5y}{dx^5} = 2520x^2]

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Question 8

Find [\dfrac{d^3y}{dx^3}] if [y = e^x\cos x].

→ Apply product rule multiple times.

Solution:

1. [\dfrac{dy}{dx} = e^x(\cos x – \sin x)]
2. [\dfrac{d^2y}{dx^2} = -2e^x\sin x]
3. [\dfrac{d^3y}{dx^3} = -3e^x\sin x – 2e^x\cos x]

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Question 9

Find [\dfrac{d^2y}{dx^2}] if [y = \tan x].

1. [\dfrac{dy}{dx} = \sec^2 x]
2. [\dfrac{d^2y}{dx^2} = 2\sec^2 x \tan x]

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Question 10

Find [\dfrac{d^2y}{dx^2}] if [y = x^3 e^x].

Steps:

1. [\dfrac{dy}{dx} ][= 3x^2 e^x + x^3 e^x]
2. [\dfrac{d^2y}{dx^2} ][= 6x e^x + 6x^2 e^x + x^3 e^x]