Question

Derivative of $(x+3{)}^2(x+4{)}^3(x+5{)}^4$ $w.r.$ to $x$ is

• A. $(x+3)(x+4)(x+5{)}^2(9x^2+70x+133)$
• B. $(x+3)(x+4{)}^2(x+5{)}^3(9x^2+70x+133)$
• C. $(x+3)(x+4{)}^2(x+5)(9x^2-70x-133)$
• D. none of these

Answer 1

The correct option is B $(x+3)(x+4{)}^2(x+5{)}^3(9x^2+70x+133)$

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

$\frac{d}{dx}\left[{(x+3)}^2{(x+4)}^3{(x+5)}^4\right]$

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

Using chain rule we solve the above expressing

$\frac{d}{dx}\left[P\left(x\right)Q\left(x\right)R\left(x\right)\right]=P^{\prime }\left(x\right)Q\left(x\right)R\left(x\right)+P\left(x\right)Q^{\prime }\left(x\right)R\left(x\right)+P\left(x\right)Q\left(x\right)R^{\prime }\left(x\right)$

$Q$ simplifying

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

$=(x+3){(x+4)}^2{(x+5)}^3[{2x}^2+18x+40+3x^2+24x+45+{4x}^2+28x+48]$

$=(x+3){(x+4)}^2{(x+5)}^3[{9x}^2+70x+133]$.