How do you differentiate $y = ln ({(6 x - 5)}^{6})$ $y=ln((6x-5)^6)$ using the chain rule?

Question

How do you differentiate $y = ln ({(6 x - 5)}^{6})$ $y=ln((6x-5)^6)$ using the chain rule?

2 Answers

Impact of this question

3226 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-06-01T17:50:00

$\frac{d}{d x} (ln ({(6 x - 5)}^{6})) = \frac{36}{6 x - 5}$ $frac{d}{dx}(ln ((6x-5)^6))=frac{36}{6x-5}$

Explanation:

$\frac{d}{d x} (ln ({(6 x - 5)}^{6}))$ $frac{d}{dx}(ln ((6x-5)^6))$

Applying chain rule, $\frac{d f (u)}{d x} = \frac{d f}{d u} \cdot \frac{d u}{d x}$ $frac{df(u)}{dx}=frac{df}{du}\cdot \frac{du}{dx}$

Let, ${(6 x - 5)}^{6} = u$ $(6x-5)^6=u$
$= \frac{d}{d u} (ln (u)) \frac{d}{d x} ({(6 x - 5)}^{6})$ $=frac{d}{du}(ln (u))frac{d}{dx}((6x-5)^6)$

We know,
$\frac{d}{d u} (ln (u)) = \frac{1}{u}$ $frac{d}{du}(ln (u))=frac{1}{u}$
and,
$\frac{d}{d x} ({(6 x - 5)}^{6}) = 36 {(6 x - 5)}^{5}$ $frac{d}{dx}((6x-5)^6)=36(6x-5)^5$

So,
$= \frac{1}{u} 36 {(6 x - 5)}^{5}$ $=\frac{1}{u}36(6x-5)^5$

substituting back, $: u = {(6 x - 5)}^{6}$ $\:u=(6x-5)^6$

$= \frac{1}{{(6 x - 5)}^{6}} 36 {(6 x - 5)}^{5}$ $=\frac{1}{(6x-5)^6}36(6x-5)^5$

Simplifying it,we get,

$\frac{36}{6 x - 5}$ $frac{36}{6x-5}$

Answer 2 · 2016-06-01T18:15:43

Use properties of logarithms to write $y = 6 ln (6 x - 5)$ $y=6ln(6x-5)$ then us $\frac{d}{d x} (ln u) = \frac{1}{u} \frac{d u}{d x}$ $d/dx(lnu) = 1/u (du)/dx$

Explanation:

$y = ln ({(6 x - 5)}^{6}) = 6 ln (6 x - 5)$ $y = ln((6x-5)^6) = 6ln(6x-5)$

So,

$\frac{d y}{d x} = 6 [\frac{1}{6 x - 5} \frac{d}{d x} (6 x - 5)]$ $dy/dx = 6[1/(6x-5) d/dx(6x-5)]$

$= 6 [\frac{1}{6 x - 5} (6)]$ $= 6[1/(6x-5) (6)]$

$= \frac{36}{6 x - 5}$ $= 36/(6x-5)$

Science

Math

Humanities

... and beyond

How do you differentiate $y = ln ({(6 x - 5)}^{6})$ $y=ln((6x-5)^6)$ using the chain rule?

2 Answers

Explanation:

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

How do you differentiate y=ln((6x−5)6) y=ln((6x-5)^6) using the chain rule?

2 Answers

Explanation:

Explanation:

Related questions

Impact of this question

How do you differentiate $y = ln ({(6 x - 5)}^{6})$ $y=ln((6x-5)^6)$ using the chain rule?