directional derivatives: NEED CLARIFICATION :(
compute the directional derivatives of f in the given directions V at the given points p f(x,y,z)=xy^2 +y^2z^3 + z^3x, p=(4,-2,-1), v=(1/ sqroot14) (i + 3j + 2k)
after you find the partial derivative of x,y, and z, i know you have to use the formula: (partial dervative of x)(v1) + (partial dervative of y)(v2) + (partial dervative of z)(v3) BUT do i just mulitple 1/sqroot14 to i, j and k OR do I assume v=(1/ sqroot14) (i + 3j + 2k) is in the (v)/(||v||) form and v1,v2,v3 is actually 1, 3, and 2 respectively?
Re: directional derivatives
Yes, you have to multiply the 1/sqrt(14) to the vector and then take the dot product of f and v. However, you should also multiply v by the reciprocal of the magnitude of v. In other words, find the normal of v.
n=1/sqrt(4^2+(-2)^2+(-1)^2) = 1/sqrt(21)
n*v =( sqrt(6)/42)(i + 3j + 2k)
Check this link with tons of examples and solutions. Go directly to page 8 and you'll see an example of your question. Also, read about DOT PRODUCT.