Question

if u=x3y4z2, and x=t2,y=t3,z=t4,then find du/dt.​

Answer 1

ANSWER:

$\sf The \ value \ of \ \dfrac{du}{dt} =26 \ {t}^{25}$

GIVEN:

• u = x³y⁴z² , x = t², y = t³, z = t⁴.

TO FIND:

• The value of du/dt.

EXPLANATION:

$\sf u = x^3y^4z^2$

$\sf \dfrac{du}{dt} = \dfrac{d}{dt} (x^3y^4z^2)$

Substitute x = t², y = t³, z = t⁴.

$\sf \dfrac{du}{dt} = \dfrac{d}{dt} (( {t}^{2})^3( {t}^{3})^4( {t}^{4})^2)$

$\sf \dfrac{du}{dt} = \dfrac{d}{dt} ({t}^{6} \times {t}^{12} \times {t}^{8})$

$\boxed{\bold{\large{\gray{x^m \times x^n = x^{m+n}}}}}$

$\sf \dfrac{du}{dt} = \dfrac{d}{dt} ({t}^{6 + 1 2+ 8} )$

$\sf \dfrac{du}{dt} = \dfrac{d}{dt} ({t}^{26} )$

$\boxed{ \bold{ \large{ \gray{ \dfrac{d}{dx} {x}^{n} = n {x}^{n - 1} }}}}$

$\sf \dfrac{du}{dt} =26 \ {t}^{26 - 1} \left(\dfrac{dt}{dt} \right)$

$\boxed{ \bold{ \large{ \gray{ \dfrac{d}{dx}x = 1 }}}}$

$\sf\dfrac{du}{dt} =26 \ {t}^{25}$

$\sf The \ value \ of \ \dfrac{du}{dt} =26 \ {t}^{25}$