Extension: for $f(x)=\frac1{3x+2}$, $x\ne-\frac23$, (a) use first principles to find $f'(x)$;

Question

Extension: for $f(x)=\frac1{3x+2}$, $x\ne-\frac23$, (a) use first principles to find $f'(x)$; (b) prove by induction that $f^{(n)}(x)=\frac{(-1)^n3^n n!}{(3x+2)^{n+1}}$ for every integer $n\ge1$.

Accepted Answer

$f'(x)=-\frac3{(3x+2)^2};\quad f^{(n)}(x)=\frac{(-1)^n3^nn!}{(3x+2)^{n+1}}$

Reciprocal derivative and induction

Reciprocal derivative and induction

Solution:

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