IB Mathematics Analysis and Approaches 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 $n\ge 1$, 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 $$f^{(n)}(x)=a^n{e}^{ax}$$