How Do You Use the Chain, Product, and Quotient Rules?

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Before differentiating, rewrite roots and reciprocal powers where useful, then work from the outside structure inward. A quotient can sometimes be rewritten as a product, but this is optional; choose the form that makes the derivative clearest. If several rules appear, apply the rule describing the whole expression first. Brackets are crucial because they show which function is outer and which is inner.

Chain Rule

Differentiate the outer function, keep the inner function unchanged, and multiply by the derivative of the inner function. For

$y=(3x^2+1)^5,$

$\frac{dy}{dx}=5(3x^2+1)^4(6x)=30x(3x^2+1)^4.$

The factor $6x$ is essential because it is the derivative of the inner function $3x^2+1$.

Product Rule

For $y=x^2e^x$, let $u=x^2$ and $v=e^x$. Then

$\frac{dy}{dx}=(2x)e^x+x^2e^x=e^x(2x+x^2).$

Do not simply multiply the derivatives: in general, $(uv)'\neq u'v'$.

Quotient Rule

For $y=\dfrac{\sin x}{x}$, let $u=\sin x$ and $v=x$. Therefore,

$\frac{dy}{dx}=\frac{x\cos x-\sin x}{x^2}.$

Remember: denominator times derivative of numerator, minus numerator times derivative of denominator, divided by the denominator squared.

These rules can be combined. For $y=x(2x+1)^4$, apply the product rule first and the chain rule within the second term:

$\frac{dy}{dx}=(2x+1)^4+8x(2x+1)^3.$

A common misconception is applying only one rule because the expression initially looks like a product or quotient. Always inspect each factor for an inner composite function.

Exam Technique

For the command term Differentiate, define $u$ and $v$ where helpful and show the substituted rule before simplifying. A GDC does not replace mathematical working, and trigonometric differentiation assumes radians. Check signs carefully in the quotient rule and never omit the inner derivative in the chain rule.