Mathematics

The key is recognizing that $\arctan$ is a composite function wrapping a product. To find $h'(x)$, apply the chain rule: differentiate the outer $\arctan$ layer (which gives $\frac{1}{1+u^2}$ where $u=x^2e^{3x}$), then multiply by the derivative of the inner function $x^2e^{3x}$ using the product rule. This layered approach—chain rule on the outside, product rule on the inside—is the standard way to handle compositions of transcendental functions.

The key is recognizing that $\arctan$ is a composite function wrapping a product. To find $h'(x)$, apply the chain rule: differentiate the outer $\arctan$ layer (which gives $\frac{1}{1+u^2}$ where $u=x^2e^{3x}$), then multiply by the derivative of the inner function $x^2e^{3x}$ using the product rule. This layered approach—chain rule on the outside, product rule on the inside—is the standard way to handle compositions of transcendental functions.