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Section 13.7 : Directional Derivatives
For problems 1 & 2 determine the gradient of the given function.
- \(\displaystyle f\left( {x,y} \right) = {x^2}\sec \left( {3x} \right) - \frac{{{x^2}}}{{{y^3}}}\) Solution
- \(f\left( {x,y,z} \right) = x\cos \left( {xy} \right) + {z^2}{y^4} - 7xz\) Solution
For problems 3 & 4 determine \({D_{\vec u}}f\) for the given function in the indicated direction.
- \(\displaystyle f\left( {x,y} \right) = \cos \left( {\frac{x}{y}} \right)\) in the direction of \(\vec v = \left\langle {3, - 4} \right\rangle \) Solution
- \(f\left( {x,y,z} \right) = {x^2}{y^3} - 4xz\) in the direction of \(\vec v = \left\langle { - 1,2,0} \right\rangle \) Solution
- Determine \({D_{\vec u}}f\left( {3, - 1,0} \right)\) for \(f\left( {x,y,z} \right) = 4x - {y^2}{{\bf{e}}^{3x\,z}}\) in the direction of \(\vec v = \left\langle { - 1,4,2} \right\rangle \). Solution
For problems 6 & 7 find the maximum rate of change of the function at the indicated point and the direction in which this maximum rate of change occurs.