Q.For a positive constant $a$, find $\frac{dy}{dx}$, where $y = a^{t + \frac{1}{t}}$, and $x = \left(t + \frac{1}{t}\right)^a$.

Both $x$ and $y$ depend on $t$, so use $\dfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt}$. Let $u=t+\frac1t$, so $\dfrac{du}{dt}=1-\dfrac1{t^2}$.

$y=a^{u}$: $\dfrac{dy}{dt}=a^{u}\log a\left(1-\dfrac1{t^2}\right)$.

$x=u^{a}$: $\dfrac{dx}{dt}=a\,u^{a-1}\left(1-\dfrac1{t^2}\right)$.

Dividing, the $\left(1-\frac1{t^2}\right)$ factor cancels: …

When $x$ and $y$ are both given through a parameter $t$, we don't eliminate $t$; we differentiate each with respect to $t$ and use

Let $u = t+\dfrac1t$, so $\dfrac{du}{dt}=1-\dfrac1{t^2}$.

1. Differentiate $y=a^{u}$

$a$ is a positive constant, so $\frac{d}{dt}a^{u}=a^{u}\log a\cdot\frac{du}{dt}$:

2. Differentiate $x=u^{a}$

This is a power with the constant exponent $a$, so $\frac{d}{dt}u^{a}=a\,u^{a-1}\cdot\frac{du}{dt}$:

3. Take the ratio

$\frac{dy}{dx}=\frac{a^{t+1/t}\log a\left(1-\frac1{t^2}\right)}{a\left(t+\frac1t\right)^{a-1}\left(1-\frac1{t^2}\right)}.$ …

Method: Parametric Differentiation with $a^{u(t)}$ and $[u(t)]^{a}$ Forms

Use this when both $x$ and $y$ are given as functions of a common parameter $t$, and one of them is a constant raised to a $t$-dependent power while the other is a $t$-dependent expression raised to a constant power.

Steps

Step 1: Set up the parametric ratio

Since neither $x$ nor $y$ is directly a function of the other, use

Step 2: Differentiate the constant-base, variable-exponent piece

For an expression of the form $a^{u(t)}$ (constant base $a$, exponent depending on $t$), use the exponential rule with the chain rule:

Step 3: Differentiate the variable-base, constant-exponent piece

For an expression of the form $[u(t)]^{a}$ (base depending on $t$, constant exponent $a$), use the ordinary power rule with the chain rule:

These two rules are easy to swap by mistake, so identify carefully which quantity is the fixed constant and which is the $t$-dependent piece before differentiating.

Step 4: Form the ratio and simplify …

Common Mistakes

Mistake 1: Swapping the differentiation rules for $a^{u}$ and $u^{a}$

Why it's wrong: $a^{u}$ has a constant base, so it differentiates as $a^{u}\log a\cdot u'$; but $u^{a}$ has a constant exponent, so it differentiates as $a\,u^{a-1}\cdot u'$. Mixing these up (e.g. using the power rule on $a^{u}$) gives a completely wrong derivative. Correct approach: before differentiating, identify which quantity is fixed (the constant) and which is the $t$-dependent piece.

Mistake 2: Stopping at $dy/dt$ instead of forming $dy/dx$

Why it's wrong: the question asks for $\frac{dy}{dx}$, but it's easy to compute $\frac{dy}{dt}$ and present that as the final answer, forgetting to divide by $\frac{dx}{dt}$. Correct approach: always finish parametric problems by explicitly writing $\frac{dy}{dx}=\frac{dy/dt}{dx/dt}$.

Mistake 3: Sign error differentiating $t+\frac1t$ …