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Fermat differentiation is a type of differentiation. The Fermat derivative was first defined by Pierre de Fermat and resulted from his work in combinatorics. The Fermat derivative is written as ${\displaystyle F_{x}[f(x)]}$ and is defined for polynomials as:

${\displaystyle F_{x}[f(x)]=f'(x)\cdot \sum _{i=1}^{d(f(x))+1}(_{f'(i)}C_{f'(i-1)}),\,}$

where ${\displaystyle d(f(x))}$ is the degree of ${\displaystyle f(x)}$.

The Fermat derivative measures the rate of change of the Fermat equation

${\displaystyle F(x)=_{f(x)}C_{f'(x)}.\,}$

For example, let ${\displaystyle f(x)=x^{2}+x+1}$. The Fermat derivative is therefore ${\displaystyle F_{x}[x^{2}+x+1]=(2x+1)\cdot (3+10+21)=68x+34}$. The corresponding Fermat equation is therefore ${\displaystyle F(x)=_{x^{2}+x+1}C_{2x+1))$. The rate of change at point ${\displaystyle (x,F(x))}$ is equal to ${\displaystyle F_{x}[f(x)]}$, which is ${\displaystyle 68x+34}$. For example, at ${\displaystyle x=1}$, the rate of change of ${\displaystyle F(x)}$ is ${\displaystyle F_{x}[f(x)]=68(1)+34=102}$.

• Calculus Abbott, P. and Neill, Hugh (Contemporary Books 2003) ISBN 9780071421287
• Advanced Calculus Demystified Bachman, (Mc-Graw Hill 2007) ISBN 9780071481212

[[Category:Differential calculus]]