When do you use the chain rule vs the product rule

Both rules arise from the limit definition of the derivative, but they apply to different structures. A product of functions looks like (u(x)v(x)); each factor varies with (x) independently. A composition looks like (g(f(x))); the inner function (f) feeds its output into the outer function (g). Recognizing the structure before you start prevents wasted algebra and sign errors.

Product Rule

The product rule states ((uv)' = u'v + uv'). For example, let (u(x)=x^2) and (v(x)=$\sin$ x). Then (u'(x)=2x) and (v'(x)=$\cos$ x). Applying the rule gives $$(x^2\sin x)' = 2x\sin x + x^2\cos x.$$ This result shows how each factor contributes a term, and the rule works no matter how complicated each factor is, as long as they are multiplied together.

Use the product rule whenever:

• You have an explicit multiplication of two differentiable functions, e.g., ($x^3$\cdot $e^x$).
• One factor is a constant multiple of another function, e.g., (5\cdot\ln x).
• The expression can be rewritten as a product after algebraic manipulation, e.g., $\frac{x}{1+x}=x\cdot(1+x)^{-1}$.

Chain Rule

The chain rule handles compositions: ((g\circ f)'(x)=g'(f(x))$\cdot$ f'(x)). Take (h(x)=(3x+2)^4). Here the outer function is (g(u)=u^4) and the inner function is (f(x)=3x+2). Compute (g'(u)=4u^3) and (f'(x)=3). Substituting gives $$\begin{aligned}h'(x) &= 4(3x+2)^3\cdot3 \\ &= 12(3x+2)^3.\end{aligned}$$ The derivative follows the “outside times the derivative of the inside” pattern.

Applying the chain rule step‑by‑step:

1. 1Identify the outer function (g) and the inner function (f).
2. 2Differentiate the outer function, keeping the inner function unchanged.
3. 3Differentiate the inner function.
4. 4Multiply the results from the previous two steps.