Prove Nth derivative expression by induction

Question

I am given the function $f(x) = \sqrt{3x+5}$

I have calculated the expression of the nth derivative to be

$$f^{(n)}(x)=\frac{(-1)^{n+1}\cdot(2n-3)!!}{2^n}\cdot(3x+5)^{-(2n-1)/2}\cdot3$$

How would I prove this expression to be true by induction?

Answer 1

You want to show that: $$f^{(n+1)}(x)=\frac{(-1)^{n}\cdot(2n-1)!!}{2^{n+1}}\cdot(3x+5)^{-(2n+1)/2}\cdot3 = \frac{d}{dx}\left(\frac{d^n}{dx^n}(f(x))\right)=\frac{d}{dx}\left(\frac{(-1)^{n+1}\cdot(2n-3)!!}{2^n}\cdot(3x+5)^{-(2n-1)/2}\cdot3\right).$$

I trust that you can differentiate the last expression.