Exercise · ZW-DIRECTIONAL-DERIVATIVE-010 · ★★★ Advanced
A differentiable function satisfies \(D_{\mathbf{i}}f(p)=2\) and \(D_{\mathbf{j}}f(p)=-1\). Compute \(D_{\mathbf{u}}f(p)\) with \(\mathbf{u}=\langle\tfrac35,\tfrac45\rangle\).
Choose one option.
A \(\tfrac25\) B \(\tfrac{14}{5}\) C \(-\tfrac25\) D \(1\) Show answer ← Back to the question
Explanation Since \(D_{\mathbf{i}}f=f_x\) and \(D_{\mathbf{j}}f=f_y\), the gradient is \(\langle2,-1\rangle\) and \(D_{\mathbf{u}}f=\tfrac65-\tfrac45=\tfrac25\).
The directional derivative \(D_{\mathbf{u}}f\) of \(f\) at a point \(\mathbf{x}_0\), along a unit vector \(\mathbf{u}\), is the rate of change of \(f\) moving from \(\mathbf{x}_0\) in the direction of \(\mathbf{u}\).
\[D_{\mathbf{u}}f(\mathbf{x}_0)=\lim_{h\to0}\frac{f(\mathbf{x}_0+h\mathbf{u})-f(\mathbf{x}_0)}{h}\]
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