[tex]f(x,y)=\dfrac{y^3}x[/tex]
a. The gradient is
[tex]\nabla f(x,y)=\dfrac{\partial f}{\partial x}\,\vec\imath+\dfrac{\partial f}{\partial y}\,\vec\jmath[/tex]
[tex]\boxed{\nabla f(x,y)=-\dfrac{y^3}{x^2}\,\vec\imath+\dfrac{3y^2}x\,\vec\jmath}[/tex]
b. The gradient at point P(1, 2) is
[tex]\boxed{\nabla f(1,2)=-8\,\vec\imath+12\,\vec\jmath}[/tex]
c. The derivative of [tex]f[/tex] at P in the direction of [tex]\vec u[/tex] is
[tex]D_{\vec u}f(1,2)=\nabla f(1,2)\cdot\dfrac{\vec u}{\|\vec u\|}[/tex]
It looks like
[tex]\vec u=\dfrac{13}2\,\vec\imath+5\,\vec\jmath[/tex]
so that
[tex]\|\vec u\|=\sqrt{\left(\dfrac{13}2\right)^2+5^2}=\dfrac{\sqrt{269}}2[/tex]
Then
[tex]D_{\vec u}f(1,2)=\dfrac{\left(-8\,\vec\imath+12\,\vec\jmath\right)\cdot\left(\frac{13}2\,\vec\imath+5\,\vec\jmath\right)}{\frac{\sqrt{269}}2}[/tex]
[tex]\boxed{D_{\vec u}f(1,2)=\dfrac{16}{\sqrt{269}}}[/tex]
