How to tell whether a composite function is increasing or decreasing at a point

Key takeaway: For a composition like $h(g(x))$, the chain rule tells you how the slope behaves. Then the sign of the derivative tells you whether the function is rising or falling.

To decide whether $y=h(g(x))$ is increasing or decreasing at $x=3$, differentiate.

Step 1: Apply the chain rule

$y' = h'(g(x))\,g'(x)$

Step 2: Evaluate at $x=3$

$y'(3)=h'(g(3))\,g'(3)$

Step 3: Interpret the sign

• If $y'(3)>0$, the function is increasing at $x=3$.
• If $y'(3)<0$, the function is decreasing at $x=3$.
• If $y'(3)=0$, it is neither increasing nor decreasing right at that point.

So the direction depends on the sign of $h'(g(3))g'(3)$.

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Pitfalls the pros know 👇 A frequent slip is to look only at $h'(x)$ or only at $g'(x)$. For a composite function, you must evaluate the outside derivative at the inside value: $h'(g(3))$, not just $h'(3)$.

What if the problem changes? If $g'(3)$ is negative, the inside function is locally decreasing. Even if $h$ is increasing, the composition can still end up decreasing because the product $h'(g(3))g'(3)$ may be negative. The sign of both factors matters.

Tags: chain rule, increasing function, decreasing function