Question Type 4: Induction with n-th order derivatives Exercises

Prove by induction that for integer $m \ge n \ge 0$, the $n$th derivative of $f(x)=(ax+b)^m$ is given by: $f^{(n)}(x)=a^n\,m(m-1)\cdots(m-n+1)\,(ax+b)^{m-n}$

By induction, derive the formula for the $n$th derivative of $f(x)=x^2 e^{3x}$.

Find the $n$th derivative of $f(x)=x^{-1}$ using mathematical induction.

Determine by induction the $n$th derivative of $f(x)=x e^{px}$ and show that $f^{(n)}(x)=p^{n-1}(px + n)e^{px}.$

Prove by induction that for $f(x)=x^m$ and integer $0 \le n \le m$, $f^{(n)}(x)=\frac{m!}{(m-n)!}\,x^{m-n}.$

Find a general expression for the $n$th derivative of $f(x)=\sin(bx)$.

Show by induction that the $n$th derivative of $f(x)=\cos(bx)$ is $f^{(n)}(x)=b^n\cos\left(bx+\frac{n\pi}{2}\right).$

Prove by induction that for $n\ge1$, the $n$th derivative of $f(x)=\ln(x)$ is $f^{(n)}(x)=(-1)^{n-1}\frac{(n-1)!}{x^n}.$

Show by induction that the $n$th derivative of $f(x)=e^{ax}$ is $f^{(n)}(x)=a^n e^{ax}$