Derivative of a composed constant-input function

Key takeaway: This one looks like a chain rule problem, but there is a small twist: the inner expression $g(3)$ is a constant, not a variable in $x$. That changes everything.

We are given

$y=f(g(3))$

and asked to find

$\frac{dy}{dx}$

Step 1: Notice what depends on $x$

The expression $g(3)$ is a constant, because the input is fixed at 3. So $f(g(3))$ is also just a constant number.

Step 2: Differentiate a constant

The derivative of a constant with respect to $x$ is

$\frac{dy}{dx}=0$

Final answer

$\boxed{0}$

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Pitfalls the pros know 👇 The biggest mistake is to apply the chain rule as if the inside were $g(x)$. It is not. Since the input is fixed at 3, there is no $x$ left in the expression, so the derivative is simply zero.

What if the problem changes? If the problem had been $y=f(g(x))$, then the chain rule would give

$\frac{dy}{dx}=f'(g(x))g'(x)$

That version is different because the inner function actually depends on $x$.

Tags: chain rule, constant function, composition