If $x(m) = am^2$, $y(m) = a / m^2$, then find the value of $\dfrac{dy}{dx}$.
(A) $\dfrac{1}{m^2}$
(B) $\dfrac{-1}{m^2}$
(C) $\dfrac{1}{m^4}$
(D) $\dfrac{-1}{m^4}$

Question

Accepted Answer

> [!TLDR]
> Parametric differentiation: $\dfrac{dy}{dx}=\dfrac{dy/dm}{dx/dm}=\dfrac{-2a/m^3}{2am}=-\dfrac{1}{m^4}$.

### Step 1 — Differentiate $x$ with respect to $m$
$$x=am^2\ \Rightarrow\ \frac{dx}{dm}=2am$$

### Step 2 — Differentiate $y$ with respect to $m$
$$y=\frac{a}{m^2}=am^{-2}\ \Rightarrow\ \frac{dy}{dm}=-2am^{-3}=-\frac{2a}{m^3}$$

### Step 3 — Form the ratio
$$\frac{dy}{dx}=\frac{dy/dm}{dx/dm}=\frac{-2a/m^3}{2am}=\frac{-2a}{m^3}\cdot\frac{1}{2am}=-\frac{1}{m^4}$$ …

Step 1 — Differentiate xxx with respect to mmm

Step 2 — Differentiate yyy with respect to mmm

Step 3 — Form the ratio

🎓Unlock everything free for 14 days

Step 1 — Differentiate $x$ with respect to $m$

Step 2 — Differentiate $y$ with respect to $m$

Unlock everything free for 14 days