Why does changing $dx$ to $dt$ change the value of $\frac{d}{dx}e^{x^2}$?

Question

First of all, I know how to correctly solve the question and get the answer which is $2xe^{x^2}$.

I wanted to ask why this approach is giving wrong results.

1. Let $x^2 = t$.
2. Then, $2xdx = dt$.
3. Which means $dx = \frac{dt}{2x} \implies dx = \frac{dt}{2t^{1/2}}$.
4. Then original equation becomes $d(e^{x^2})/dx \implies d(e^t)\cdot2\cdot(t^{1/2})/dt$
5. This differentiation gives result as: $\frac{(2t+1)e^t}{2t^{1/2}}$.
6. Which when translated in $x$ gives $\frac{(2x^2+1)e^{x^2}}{2x}$

Why is this wrong? $\frac{(2x^2+1)e^{x^2}}{2x}$.

Answer 1

You have misapplied the chain rule.

Start with $e^{x^2}$

Make the substitution. $x^2 = t$

$e^t$

Do not apply $\frac {dt}{dx}$ until you have differentiated.

$\frac {d}{dx} e^{x^2} = \frac {d}{dt} e^t \frac {dt}{dx}\\ 2x e^{x^2} = (e^t)(2t^\frac 12)$

Answer 2

Two errors occurred when going from step $4$ to step $5$.

First, the correct expression to simplify has only the $e^t$ inside the differential in the numerator. So, this should result in $\frac{e^t dt \cdot 2 \cdot t^{1/2}}{dt}=2 e^t t^{1/2}=2e^{x^2} (x^2)^{1/2}=2x e^{x^2}$.

Second, not only did you simplify the wrong expression, but even worse, you did not even get the correct answer for the wrong expression. The answer for $\frac{d(e^t \cdot 2 \cdot t^{1/2})}{dt}$ is actually $\frac{(2t+1) e^t}{t^{1/2}}$ with no factor of $2$ in the denominator.