Question

Given that z= x² + y² + e^t, y=sin(x), x= cos(t). Find dz/dt

Answer 1

Required Formulae:

$(1)\ \frac{d}{dx}\sin (x)=\cos(x)\\ (2) \ \frac{d}{dx}\cos(x)=-\sin(x)\\ (3) \ \frac{d}{dx}e^x=e^x$

(4) Chain rule of derivative: $\frac{du}{dv}=\frac{du}{dx}\frac{dx}{dv}$

Solution:

Take

$x=\cos(t)\\ \Rightarrow \frac{dx}{dt}=-\sin(t)$

Now, take

$y=\sin(x)\\ \Rightarrow \frac{dy}{dt}=\frac{d}{dt}\sin( x)=\frac{d}{dx}\sin (x).\frac{dx}{dt}=-\cos(x).\sin(t)$

Finally,

$z=x^2+y^2+e^t\\ \Rightarrow \frac{dz}{dt}=\frac{d}{dt}(x^2)+\frac{d}{dt}(y^2)+\frac{d}{dt}(e^t)\\ \Rightarrow \frac{dz}{dt}=\frac{d}{dx}(x^2).\frac{dx}{dt}+\frac{d}{dy}(y^2).\frac{dy}{dt}+\frac{d}{dt}(e^t)\\ \Rightarrow \frac{dz}{dt}=2x.\frac{dx}{dt}+2y.\frac{dy}{dt}+e^t \\ \Rightarrow \boxed{\frac{dz}{dt}=-2x.\sin(t)-2y.\cos(x).\sin(t)+e^t}$