Calculus
Topics

# How do you find the derivative of f(x) = (x^2-4)/(x-1)?

Jun 6, 2016

In a function $f \left(x\right) = \frac{g \left(x\right)}{h \left(x\right)}$, the derivative is given by $\frac{\left(g ' \left(x\right) \times h \left(x\right)\right) - \left(g \left(x\right) \times h ' \left(x\right)\right)}{h \left(x\right)} ^ 2$

#### Explanation:

We must determine the derivatives of the numerator and of the denominator,

Numerator:

Let $g \left(x\right) = {x}^{2} - 4$

By the power rule, $g ' \left(x\right) = 2 x$.

Denominator:

Let $h \left(x\right) = x - 1$

By the power rule, $h ' \left(x\right) = 1$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
We can now differentiate with the quotient rule.

$f ' \left(x\right) = \frac{\left(g ' \left(x\right) \times h \left(x\right)\right) - \left(g \left(x\right) \times h ' \left(x\right)\right)}{h \left(x\right)} ^ 2$

$f ' \left(x\right) = \frac{\left(2 x \left(x - 1\right)\right) - \left(\left({x}^{2} - 4\right) 1\right)}{x - 1} ^ 2$

$f ' \left(x\right) = \frac{2 {x}^{2} - 2 x - {x}^{2} + 4}{x - 1} ^ 2$

$f ' \left(x\right) = \frac{{x}^{2} - 2 x + 4}{{x}^{2} - 2 x + 1}$

Thus, the derivative of $f \left(x\right) = \frac{{x}^{2} - 4}{x - 1}$ is $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{{x}^{2} - 2 x + 4}{{x}^{2} - 2 x + 1}$

Hopefully this helps!

##### Impact of this question
369 views around the world
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 12, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "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, "math_score": 0.9494115114212036, "perplexity": 1328.2487036760451}, "config": {"markdown_headings": true, "markdown_code": false, "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": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703549416.62/warc/CC-MAIN-20210124141945-20210124171945-00215.warc.gz"}