Question

Differentiate the given functions w.r.t. x.

$(x+3{)}^2(x+4{)}^3(x+5{)}^4$

Answer 1

Ley y = $(x+3{)}^2(x+4{)}^3(x+5{)}^4$

Taking log on both sides, we get

$logy=log\left\{\right(x+3{)}^2(x+4{)}^3(x+5{)}^4\}orlogy=2log(x+3)+3log(x+4)+4log(x+5\left)\right(\therefore log\left(lmn\right)=logl+logm+logn)$

Differentiating both sides w.r.t. x, we get

$\frac{d}{dx}\left(logy\right)=2\frac{d}{dx}log(x+3)+3\frac{d}{dx}log(x+4)+4\frac{d}{dx}log(x+5)\frac{1}{y}\frac{dy}{dx}=2\left(\frac{1}{x+3}\right)(1=0)+3\left(\frac{1}{x+4}\right)(1+0)+4\left(\frac{1}{x+5}\right)(1+0)\Rightarrow \frac{dy}{dx}=y\{\left(\frac{2}{x+3}\right)+\left(\frac{3}{x+4}\right)+\left(\frac{4}{x+5}\right)\}\Rightarrow \frac{dy}{dx}=(x+3{)}^2(x+4{)}^3(x+5{)}^4\{\left(\frac{2}{x+3}\right)+\left(\frac{3}{x+4}\right)+\left(\frac{4}{x+5}\right)\}=(x+3{)}^2(x+4{)}^3(x+5{)}^4\left\{\frac{2(x+4)(x+5)+3(x+3)(x+5)+4(x+3)(x+4)}{(x+3)(x+4)(x+5)}\right\}=(x+3)(x+4{)}^2(x+5{)}^4\left[2\right(x^2+9x+20)+3(x^2+8x+15)+4(x^2+7x+12\left)\right]=(x+3)(x+4{)}^2(x+5{)}^4(9x^2+70x+133)$