<p>Can someone explain to me how to find the partial derivative with respect to z of the following:</p>
<p>f(x,y,z)=xze^(yz)</p>
<p>thanks.</p>
<p>=xzy<em>e^(yz) + x</em>e^(yz) </p>
<p>=x*e^(yz) * (zy + 1)</p>
<p>To find the partial derivative of the function f(x,y,z) with respect to variable z, we regard the ordinary derivative of the function f(x,y,z) with respect to variable z, keeping all other variables x,y constant. Besides, we apply the product differentiation rule with respect to variable z:
(h(z)<em>g(z))’=h(z)</em>(g(z))’+ g(z)<em>(h(z))’.
So, (xz</em>e^(yz))’z=x* e^(yz)* (z)’z+xz<em>(e^(yz))’z= x</em> e^(yz)+xzy<em>e^(yz)=(x+xzy)</em> e^(yz)=x<em>(1+zy)</em> e^(yz).
Answer: x<em>(1+zy)</em> e^(yz).</p>
<p>Yes, just treat x and y as constants and differentiate as you would a single-variable function in z.</p>