Use the directional derivative to estimate the change in $f(x,y)=\cos\pi xy+xy^2$ as point $(-1,-1)$ moves distance of $0.1$ along vector $i+j$

Question

Use the directional derivative to estimate how much $f(x,y)=\cos\pi xy+xy^2$ will change if the point $P$ is moved from $(-1,-1)$ a distance of $0.1$ along the vector $i+j$

I don't know how to solve this problem can someone please give me some hints about what the problem wants?

the answer in my textbook is $0.15\sqrt2$

Answer 1

We have $$\nabla f=(-\pi y\sin\pi xy+y^2,-\pi x\sin\pi xy+2xy)$$ $$(\nabla f)(-1,-1)=(1,2)$$ The vector of length $0.1$ in the stated direction is $\frac{0.1}{\sqrt2}(1,1)$. The estimated change is then the directional derivative computed with this displacement vector, i.e. the dot product with $\nabla f$: $$\frac{0.1}{\sqrt2}(1,2)\cdot(1,1)=\frac{0.3}{\sqrt2}=\frac{0.3\sqrt2}2=0.15\sqrt2$$