# What is derivative of f(x) = 2^(arctg x)?

sciencesolve | Certified Educator

You may notice that the given function is the result of composition of two functions, g(x) = 2^x and h(x) = arctan x, such that:

(goh)(x) = g(h(x)) => g(h(x)) = 2^(h(x)) => g(h(x)) = 2^(arctan x)

Since the function `f(x)` is the result of composition of two functions, you may evaluate its derivative using the chain rule, hence, you need to start differentiation process from outside toward inside, such that:

`f'(x) = (2^(arctan x))'*(arctan x)'`

`f'(x) = (2^(arctan x)*ln 2)*(1/(1 + x^2))`

`f'(x) = (2^(arctan x)*ln 2)/(1 + x^2)`

You may also use, as alternative method, the logarithmic differentiation, such that:

`f(x) = y => y = 2^(arctan x)`

Taking common logarithms both sides, yields:

`ln y = ln (2^(arctan x))`

Using the power property of logarithms yields:

`ln y = arctan x*ln 2`

Differentiation both sides, yields:

`(1/y)*y' = (1/(1 + x^2))*ln 2 `

`y' = y*(ln 2/(1 + x^2))`

Replacing `2^(arctan x)` for y yields:

`y' = (2^(arctan x)*ln 2)/(1 + x^2)`

Hence, evaluating the derivative of the given function, using either the chain rule, or logarithmic differentiation, yields `y' = (2^(arctan x)*ln 2)/(1 + x^2).`
{"found_math": false, "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": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://www.enotes.com/homework-help/what-derivative-f-x-2-arctg-x-447515", "fetch_time": 1521306042000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2018-13/segments/1521257645248.22/warc/CC-MAIN-20180317155348-20180317175348-00317.warc.gz", "warc_record_offset": 812993474, "warc_record_length": 9691, "token_count": 422, "char_count": 1220, "score": 4.25, "int_score": 4, "crawl": "CC-MAIN-2018-13", "snapshot_type": "latest", "language": "en", "language_score": 0.8273672461509705, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00009-of-00064.parquet"}