Question

Find the indicated derivative.$$\frac{d^{5}}{d x^{5}}\left[a x^{4}+b x^{3}+c x^{2}+d x+e\right]$$.

Name: Problem 42, Chapter 3: Calculus: One Variable
Uploaded: 2021-07-28T11:05:42-08:00
Duration: 2 min 2 s
Channel: Matthys Marthinus
Description: Find the indicated derivative.(d^5)/(d x^5)[a x^4+b x^3+c x^2+d x+e].

   Find the indicated derivative.$$\frac{d^{5}}{d x^{5}}\left[a x^{4}+b x^{3}+c x^{2}+d x+e\right]$$.

Calculus: One Variable

Garret J. Etgen,… 10th Edition

Chapter 3, Problem 42

Instant Answer

Solved by Expert Matthys Marthinus

07/28/2021

Step 1

Step 1: First, we take the first derivative of the function: \[\frac{d}{dx}\left[a x^{4}+b x^{3}+c x^{2}+d x+e\right] = 4a x^{3}+3b x^{2}+2c x+d\] Here, the constant term 'e' disappears as the derivative of a constant is zero. Show more…

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Find the indicated derivative.$$\frac{d^{5}}{d x^{5}}\left[a x^{4}+b x^{3}+c x^{2}+d x+e\right]$$.

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Key Concepts

Nth Derivative

The nth derivative of a function is obtained by differentiating the function repeatedly n times. In the context of polynomials, each differentiation reduces the degree of the polynomial, and if the number of differentiations exceeds the degree of the polynomial, the result will be zero.

Polynomial Differentiation

Differentiation of a polynomial function involves applying the power rule to each term, where the derivative of ax^n is n·a·x^(n?1). This systematic reduction in degree makes it straightforward to compute higher-order derivatives until the derivative becomes identically zero.

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