write an expression for the length of the rope from the top of the second pole to the ground l2 using only…
write an expression for the length of the rope from the top of the second pole to the ground l2 using only d, m, n, and x. l2 = √(n²+(d - x)²) the total length of rope is l = l1+l2. now find dl/dx = dl1/dx + dl2/dx. differentiate l1. dl1/dx = x/√(m²+x²) differentiate l2. dl2/dx = □
Answer
Explanation:
Step1: Recall chain - rule of differentiation
If $y = \sqrt{u}$, then $\frac{dy}{dx}=\frac{1}{2\sqrt{u}}\cdot\frac{du}{dx}$. Let $u = n^{2}+(d - x)^{2}$.
Step2: Differentiate $u$ with respect to $x$
$\frac{du}{dx}=2(d - x)\cdot(- 1)=-2(d - x)$
Step3: Differentiate $L_2=\sqrt{n^{2}+(d - x)^{2}}$ using chain - rule
$\frac{dL_2}{dx}=\frac{-2(d - x)}{2\sqrt{n^{2}+(d - x)^{2}}}=\frac{x - d}{\sqrt{n^{2}+(d - x)^{2}}}$
Answer:
$\frac{x - d}{\sqrt{n^{2}+(d - x)^{2}}}$