Question

Differentiate the function with respect to $x$.
$(x+3)^2.(x+4)^3.(x+5)^4$

Answer 1

The correct answer is $∴\frac{dy}{dx}=(x+3)(x+4)^2(x+5)^3(9x^2+70x+133)$
Let $y=(x+3)^2.(x+4)^3.(x+5)^4$
Taking logarithm on both the sides,we obtain
$log\,y=log(x+3)^2+log(x+4)^3+log(x+5)^4$
$⇒log\,y=2log(x+3)+3log(x+4)+4log(x+5)$
Differentiating both sides with respect to x,we obtain
$\frac{1}{y}.\frac{dy}{dx}=2.\frac{1}{x+3}.\frac{d}{dx}(x+3)+3.\frac{1}{x+4}.\frac{d}{dx}(x+4)+4.\frac{1}{x+5}.\frac{d}{dx}(x+5)$
$⇒\frac{dy}{dx}=y[\frac{2}{x+3}+\frac{3}{x+4}+\frac{4}{x+5}]$
$⇒\frac{dy}{dx}=(x+3)^2(x+4)^3(x+5)^4.[\frac{2}{x+3}+\frac{3}{x+4}+\frac{4}{x+5}]$
$⇒\frac{dy}{dx}=(x+3)^2(x+4)^3(x+5)^4.[\frac{2(x+4)(x+5)+3(x+3)(x+5)+4(x+3)(x+4)}{(x+3)(x+4)(x+5)}]$
$\frac{dy}{dx}=(x+3)(x+4)^2(x+5)^3.[2(x^2+9x+20)+3(x^2+8x+15)+4(x^2+7x+12)]$
$∴\frac{dy}{dx}=(x+3)(x+4)^2(x+5)^3(9x^2+70x+133)$