Chain rule

Chain rule formula: h'(x) = z'(y(x)) * y'(x)

The chain rule formula connects the derivative of a composite function to the derivatives of its constituent functions. It is essential for differentiating compositions of functions in calculus. The formula states that the derivative of the outer function z with respect to y, multiplied by the derivative of y with respect to x, yields the derivative of the composite function h with respect to x.

Example

Let z(u) = u^2 and y(x) = 3x + 1. Then h(x) = z(y(x)) = (3x + 1)^2. To find h'(x), we first find z'(u) = 2u and y'(x) = 3. Then, h'(x) = z'(y(x)) * y'(x) = 2(3x + 1) * 3 = 6(3x + 1).

Understanding the chain rule formula is crucial for solving complex differentiation problems involving composite functions in calculus.

Related concepts

Lagrangian L(x,λ) = f(x) - λg(x)

L(x,λ) = f(x) - λ(g(x) - c)

Cross-entropy

Cross-entropy loss equation: H(p, q) = -Σ(p(x) * log(q(x)))

Entropy (information theory)

H(X) = −∑x∈X p(x) log(p(x))

Expected value

Expected value formula: E[X] = Σ [x * P(x)]

Covariance matrix

Covariance formula: Cov(X, Y) = E[(X - E[X])(Y - E[Y])]

Mutual information

Mutual information formula: I(X;Y) = ∑_x∈X ∑_y∈Y p(x,y) log(p(x,y)/(p(x)p(y)))

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