Determine an expression using binomial theorem

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SUMMARY

The discussion focuses on determining the expression for the polynomial function f(x) = (1+x)(1+2x)(1+3x)…(1+nx) and finding its derivative at zero, denoted as f'(0). A key insight provided is that to compute f'(0), one must identify the coefficient of x in the expanded form of f(x). This approach simplifies the problem by reducing the need for full polynomial expansion.

PREREQUISITES
  • Understanding of polynomial functions
  • Knowledge of derivatives and their applications
  • Familiarity with the binomial theorem
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study the binomial theorem and its applications in polynomial expansion
  • Learn about calculating derivatives of polynomial functions
  • Explore methods for finding coefficients in polynomial expressions
  • Investigate combinatorial interpretations of polynomial products
USEFUL FOR

Mathematicians, students studying calculus, and anyone interested in polynomial functions and their derivatives will benefit from this discussion.

Ernie1
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Determine an expression for f(x) =(1+x)(1+2x)(1+3x)…(1 +nx),find f⸍(0) .
 
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Ernie said:
Determine an expression for f(x) =(1+x)(1+2x)(1+3x)…(1 +nx),find f⸍(0) .
Hi Ernie, and welcome to MHB!

Here's a hint that may help you. If you have a polynomial $f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$, then $f'(0) = a_1$. In other words, to find $f'(0)$ you only need to find the coefficient of $x$ in $f(x)$.
 

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