Chain Rule

Definition

The chain rule gives the derivative of a composite function $y=f(g(x))$.

Intuition

“Derivative of the outside times derivative of the inside.” If $g$ stretches or compresses its input by a factor $g^{\prime }(x)$, and $f$ further scales by $f^{\prime }(g(x))$, the net rate of change is the product of these two local scaling factors.

Formal Description

If $y=y(x(t))$, the derivative of $y$ with respect to $t$ is:

$$\frac{dy}{dt}=\frac{dy}{dx}\frac{dx}{dt},$$

treating the Leibniz derivatives as ratios of differentials.

In terms of function composition:

$$[f(g(x)){]}^{\prime }=f^{\prime }(g(x))g^{\prime }(x).$$ $$\frac{d}{dx}f(g(x))=f^{\prime }(g(x))\cdot g^{\prime }(x).$$

Applications

• Differentiating composite functions such as $e^{x^2}$, $\sin (3x)$, $\ln (\cos x)$.
• Underlies implicit differentiation and the general power rule for real exponents.
• Extends to the multivariable chain rule for partial derivatives.

Trade-offs

The rule requires $g$ to be differentiable at $x$ and $f$ to be differentiable at $g(x)$. In the multivariable setting the scalar product $g^{\prime }(x)$ becomes a Jacobian matrix product.

Links

• Differentiation Rules
• Partial Derivatives
• Common Derivatives