3. Techniques of Differentiation

Higher Order Derivatives

3. Techniques of Differentiation

Higher Order Derivatives: Videos & Practice Problems

Topic summary

The second derivative of a polynomial function is found by taking the derivative twice, which is a form of higher order derivative. For example, if the first derivative is $f' (x) = 6 x - 2$ , the second derivative is $f'' (x) = 6$ , and the third derivative is $f''' (x) = 0$ . Higher order derivatives can be denoted using prime notation or by indicating the order in parentheses.

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Higher Order Derivatives

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Higher Order Derivatives Video Summary

Understanding higher order derivatives is essential in calculus, as they provide insights into the behavior of functions beyond their first derivative. The second derivative, denoted as $ f''(x) $, is simply the derivative of the first derivative, indicating how the rate of change of a function itself changes. To find the second derivative, you take the derivative of the function twice.

For example, consider the polynomial function $ f(x) = 3x^2 - 2x + 5 $. The first derivative, using the power rule, is calculated as follows:

$ f'(x) = \frac{d}{dx}(3x^2) - \frac{d}{dx}(2x) + \frac{d}{dx}(5) = 6x - 2 $

Next, to find the second derivative, we differentiate $ f'(x) $:

$ f''(x) = \frac{d}{dx}(6x - 2) = 6 $

Continuing this process, the third derivative $ f'''(x) $ is obtained by differentiating the second derivative:

$ f'''(x) = \frac{d}{dx}(6) = 0 $

At this point, the fourth derivative $ f^{(4)}(x) $ is also 0, as the derivative of a constant is zero. This pattern illustrates that once a polynomial is differentiated enough times, it will eventually yield a constant or zero.

Higher order derivatives can be denoted in various ways. The notation $ f^{(n)}(x) $ indicates the $ n $-th derivative, while $ D^n f $ or $ D_n f $ can also be used to represent the same concept. Understanding these notations and the process of repeated differentiation is crucial for analyzing the properties of functions, such as concavity and inflection points.

In summary, higher order derivatives are simply the result of taking derivatives multiple times, and they play a significant role in understanding the dynamics of functions in calculus.

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Higher Order Derivatives Example 1

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Higher Order Derivatives Example 1 Video Summary

To find the second derivative of the function $ f(x) = 4x^{-3} + 3x^{7} $, we begin by calculating the first derivative, $ f'(x) $, using the power rule. The power rule states that if $ f(x) = ax^n $, then $ f'(x) = nax^{n-1} $.

For the first term $ 4x^{-3} $, applying the power rule gives:

$ f'(x) = -3 \cdot 4x^{-3-1} = -12x^{-4} $.

For the second term $ 3x^{7} $, we again apply the power rule:

$ f'(x) = 7 \cdot 3x^{7-1} = 21x^{6} $.

Combining these results, the first derivative is:

$ f'(x) = -12x^{-4} + 21x^{6} $.

Next, we find the second derivative $ f''(x) $ by differentiating $ f'(x) $ again. We apply the power rule to each term:

For the term $ -12x^{-4} $:

$ f''(x) = -4 \cdot (-12)x^{-4-1} = 48x^{-5} $.

For the term $ 21x^{6} $:

$ f''(x) = 6 \cdot 21x^{6-1} = 126x^{5} $.

Thus, the second derivative is:

$ f''(x) = 48x^{-5} + 126x^{5} $.

This result provides insight into the concavity and behavior of the original function $ f(x) $ at various points, which is essential for understanding its graphical representation and applications in optimization problems.

example

Higher Order Derivatives Example 2

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Higher Order Derivatives Example 2 Video Summary

To find the fifth derivative of the function $ f(x) = x^8 $, we will apply the power rule repeatedly. The power rule states that if $ f(x) = x^n $, then the first derivative $ f'(x) = n \cdot x^{n-1} $.

Starting with the first derivative:

$ f'(x) = 8x^{8-1} = 8x^7 $

Next, we find the second derivative:

$ f''(x) = 7 \cdot 8x^{7-1} = 56x^6 $

For the third derivative:

$ f'''(x) = 6 \cdot 56x^{6-1} = 336x^5 $

Now, we calculate the fourth derivative:

$ f^{(4)}(x) = 5 \cdot 336x^{5-1} = 1680x^4 $

Finally, we find the fifth derivative:

$ f^{(5)}(x) = 4 \cdot 1680x^{4-1} = 6720x^3 $

Thus, the fifth derivative of the function $ f(x) = x^8 $ is:

$ f^{(5)}(x) = 6720x^3 $

This process illustrates how the power rule simplifies the differentiation of polynomial functions, allowing us to systematically reduce the exponent while multiplying by the current exponent at each step.

Problem

Find the third derivative of the given function.
$f (x) = 24 + 4 x^{5}$

$60 x squared$

$80 x cubed$

$240 x squared$

$24 minus 240 x squared$

Problem

Find the third derivative of the given function.
$y = 3 x^{2} + 9 x + 1$

$0$

$6$

$x$

$3 x squared$

Problem

Find the third derivative of the given function.
$f(t)=5t^{-4}$

$0$

$120t$

$600t^7$

$- 600 t^{- 7}$

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Higher Order Derivatives

3. Techniques of Differentiation

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3. Techniques of Differentiation

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What is a higher order derivative?

A higher order derivative is the result of differentiating a function multiple times. For example, the first derivative of a function f(x) is denoted as f'(x), the second derivative as f''(x), and so on. Higher order derivatives provide information about the curvature and concavity of the function, and they are useful in various applications such as physics and engineering. They can be denoted using prime notation or by indicating the order in parentheses, such as f⁽ⁿ⁾(x) for the nth derivative.

How do you find the second derivative of a polynomial function?

To find the second derivative of a polynomial function, you need to differentiate the function twice. For example, if you have a polynomial function f(x) = 3x³ - 2x² + 5x - 4, first find the first derivative: f'(x) = 9x² - 4x + 5. Then, differentiate the first derivative to get the second derivative: f''(x) = 18x - 4. This process involves applying the power rule and other differentiation rules repeatedly.

What is the notation for higher order derivatives?

Higher order derivatives can be denoted using several notations. The most common is the prime notation, where the first derivative is f'(x), the second derivative is f''(x), and the third derivative is f'''(x). For derivatives beyond the third, it is common to use a number in parentheses, such as f⁽⁴⁾(x) for the fourth derivative. Another notation involves using the capital letter D, such as Dⁿf(x) for the nth derivative.

How do you find the 100th derivative of a polynomial function?

Finding the 100th derivative of a polynomial function involves differentiating the function 100 times. However, in practice, this is often simplified by recognizing that higher order derivatives of polynomial functions eventually become zero. For example, if you have a polynomial function f(x) = 3x³ - 2x² + 5x - 4, the first few derivatives are: f'(x) = 9x² - 4x + 5, f''(x) = 18x - 4, f'''(x) = 18, and f⁽⁴⁾(x) = 0. Since the fourth derivative is zero, all higher order derivatives, including the 100th, will also be zero.

What is the significance of higher order derivatives in calculus?

Higher order derivatives are significant in calculus because they provide deeper insights into the behavior of functions. The first derivative gives the rate of change or slope of the function, the second derivative indicates the concavity and points of inflection, and higher order derivatives can reveal more complex aspects of the function's behavior. They are essential in fields such as physics, engineering, and economics for modeling and analyzing dynamic systems, optimizing functions, and solving differential equations.