Правило произведения

Why products need their own rule

• Tempting but wrong: the derivative of a product is not the product of the derivatives.
• $\dfrac{d}{dx}[x\cdot x]=\dfrac{d}{dx}[x^2]=2x$, but $f'g'=1\cdot1=1$. They disagree.
• Products get their own tool: the Product Rule 乘积法则.
• It accounts for both factors changing at once.

The rule

• For a product of two functions:
• $$\frac{d}{dx}\big[f\,g\big]=f'\,g+f\,g'$$
• In words: "derivative of the first times the second, plus the first times derivative of the second."
• Each term differentiates one factor and leaves the other alone; then you add.

The product rule as a growing rectangle: a top strip plus a side strip

A clean worked pattern

• Differentiate $y=x^2\,e^x$. Let $f=x^2$ and $g=e^x$.
• $f'=2x$, $g'=e^x$.
• $y'=f'g+fg'=2x\,e^x+x^2\,e^x$.
• Factor if asked: $y'=x e^x(2+x)$.

When you actually need it

• If a product can be multiplied out first, do that — it may be easier ($x^2\cdot x^3=x^5\Rightarrow5x^4$).
• But when the factors won't simplify (like $x^2\sin x$ or $e^x\ln x$), the rule is required.
• Combine it freely with the power and elementary derivatives for each factor.
• Label $f,f',g,g'$ first — organized bookkeeping prevents mistakes.