Forgetting the inner derivative in the chain rule

The chain rule differentiates a function of a function: an outer operation wrapped around an inner expression. For a power like $(3x^{2}+1)^{4}$, the outer job is the fourth power and the inner expression is $3x^{2}+1$.

The rule says bring the power down, reduce it by one, and then multiply by the derivative of the inside. The tempting slip in Add Math is to do the first part perfectly, $4(3x^{2}+1)^{3}$, and stop there, forgetting the $\times 6x$.

It happens because the power-rule half feels like the whole answer, and the inside looks like it was already dealt with when you copied it down. But without the inner derivative the working silently ignores how fast the inside itself is changing.

When the inside is just $x$ the missing factor is $1$ and no harm is done, which is exactly why the habit of leaving it off survives, until the inside is anything more than $x$, and every value that follows is wrong.

Take $y=(3x^{2}+1)^{4}$. Bringing the power down but forgetting the inside gives:

The inner expression is $3x^{2}+1$, whose derivative is $6x$. The chain rule multiplies by it:

The wrong line is missing the whole factor $6x$, so it is off by that much at every value of $x$ except where $6x=1$. A quick way to notice: after a chain-rule power, your answer should almost always carry an extra term out front that came from the inside.

1. 1

   Name the inside

   Let $u$ be the inner expression, so $y$ becomes a simple power of $u$, e.g. $y=u^{4}$ with $u=3x^{2}+1$.

2. 2

   Differentiate the outside

   Find $\frac{dy}{du}$, treating $u$ as a single block: $\frac{dy}{du}=4u^{3}$.

3. 3

   Differentiate the inside

   Find $\frac{du}{dx}$ separately: $\frac{du}{dx}=6x$. This is the factor that gets forgotten.

4. 4

   Multiply, never add

   Combine as $\frac{dy}{du}\cdot\frac{du}{dx}$, then rewrite $u$ back in terms of $x$.

5. 5

   Check the front factor

   Confirm your answer carries the inner derivative as a factor. If the inside was not simply $x$, that factor must be there.

Our teachers make the inner derivative a fixed second step, not an optional extra: name the inside, differentiate the outside, then write $\times\frac{du}{dx}$ on the same line before simplifying. In a one-to-one lesson we watch for the exact moment your pen wants to stop after the power, and we build the reflex to ask "derivative of the inside?"

every time. Because the paper awards method marks, showing $\frac{dy}{du}$ and $\frac{du}{dx}$ separately keeps your working easy to credit.

Teachers at spmaddmath.com.my are experienced, and lessons are online and taught in English. To lock in the chain rule, message us on WhatsApp to arrange a one-hour paid class from RM50/hr at the teacher's rate.