Question

Differentiate the function w.r.t. $x$.
$(x+3{)}^2.(x+4{)}^3.(x+5{)}^4$

Answer 1

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