# Thread:

1. ## nth Derivative in Proof by Induction

Hey, I've been working through some proof by induction problems, and most of them are manageable, but then I saw one which involved differentiation which I have no idea what to do. Could someone please lead me in the right direction?

$\displaystyle \mbox{Prove that the nth derivative of } y=ln(x), x>0 \mbox{ is: }<br /> \frac{d^ny}{dx^n} = \frac{(-1)^{n+1}(n-1)!}{x^n}$

I have the basic steps done:

$===================$

Step 1: Prove for n = 1

$\dfrac{dy}{dx} [ln(x)] = \dfrac{1}{x}$

$--------$

$\dfrac{d^1y}{dx^1} = \dfrac{(-1)^2(1-1)!}{x^1}$

$\dfrac{d^1y}{dx^1} = \dfrac{1}{x} = \dfrac{dy}{dx} [ln(x)] \mbox{ \therefore True }$

$===================$

Step 2: Assume for n = k

$\dfrac{d^ky}{dx^k} = \dfrac{(-1)^{k+1}(k-1)!}{x^k}$

$===================$

Step 3: Prove for n = k+1

I know how to start this, but how do I continue?

Thanks in advance for help. BG

2. For

$<br /> n=k+1<br />$

$<br /> \dfrac{d}{dx} \; \dfrac{d^ky}{dx^k} <br />$
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 11, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9760695099830627, "perplexity_flag": "middle"}