Chain, product and quotient rules

Japanese school year: Math III

What you learn

This topic teaches differentiation rules for product, quotient, and composite functions using the chain rule. These techniques are vital for computing derivatives of intricate formulas arising in physics and economic sensitivity analysis. A firm grasp of differentiating elementary polynomial and transcendental functions is essential prior knowledge.

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Key points

This product rule is used to differentiate the product of two functions, $f(x)$ and $g(x)$ . Differentiate one function while keeping the other unchanged, and add the two resulting terms together.

\{f(x)g(x)\}' = f'(x)g(x) + f(x)g'(x)

This quotient rule differentiates a fraction $\frac{f(x)}{g(x)}$ . Square the denominator to get $\{g(x)\}^2$ , and on top subtract the numerator times the denominator's derivative from the numerator's derivative times the denominator.

\left\{\frac{f(x)}{g(x)}\right\}' = \frac{f'(x)g(x) - f(x)g'(x)}{\{g(x)\}^2}

This chain rule is used to differentiate a composite function $f(g(x))$ . First differentiate the outer function $f$ , then multiply by the derivative of the inner function $g'(x)$ .

\{f(g(x))\}' = f'(g(x))g'(x)

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Practice 1Coming soon