Find the nth derivative of each function by calculating the first few derivatives ....
a)f(x)=x^n
f^(n) (x)=______
b) f(x)=1/x
f^(n) (x)=_______
+118654 Find the nth derivative of each function by calculating the first few derivatives ....
\(\begin{align}a)\quad f(x)&=x^n\\ f'^{(1)}(x)&=nx^{n-1}\\ f'^{(2)}(x)&=n(n-1)x^{n-2}\\ ...\\ f'^{(n)}(x)&=n(n-1)......1x^{n-n}\\ f'^{(n)}(x)&=n!\\ \end{align}\)
\(\begin{array} \\ b)\\\; f(x)&=&x^{-1}\\ f'^{(1)}(x)&=&-1x^{-2}\\ f'^{(2)}(x)&=&(-1*-2)x^{-3}\\ f'^{(3)}(x)&=&(-1*-2*-3)x^{-4}\\ ...\\ f'^{(n)}(x)&=&(-1*-2*-3*.....*-n)x^{-(n+1)}\\ f'^{(n)}(x)&=&(-1)^n n!\;x^{-(n+1)}\\ f'^{(n)}(x)&=&\frac{(-1)^n n!\;}{x^{(n+1)}}\\ \end{array}\)
You had best be very careful to check through my working and my logic but this answer seems ok to me.
