Question

Find the derivative of the function at P 0 in the direction of A. ​f(x,y,z) = 3 e^x cos(yz)​, P0 (0, 0, 0), A = - i + 2 j + 3k

Answer 1

The derivative of
`f(x,y,z)` at a point
`p_0=(x_0,y_0,z_0)` in the direction of a vector
`\vec a=a_x\,\vec\imath+a_y\,\vec\jmath+a_z\,\vec k` is

`\\abla f(x_0,y_0,z_0)\cdot(\vec a)/(\|\vec a\|)`

We have

`f(x,y,z)=3e^x\cos(yz)\implies\\abla f(x,y,z)=3e^x\cos(yz)\,\vec\imath-3ze^x\sin(yz)\,\vec\jmath-3ye^x\sin(yz)\,\vec k`

and

`\vec a=-\vec\imath+2\,\vec\jmath+3\,\vec k\implies\|\vec a\|=√((-1)^2+2^2+3^2)=√(14)`

Then the derivative at
`p_0` in the direction of
`\vec a` is

`3\,\vec\imath\cdot(-\vec\imath+2\,\vec\jmath+3\,\vec k)/(√(14))=-\frac3{√(14)}`