Chain Rule in Calculus

These flashcards cover key concepts and theorems related to the Chain Rule and its application in calculus.

Chain Rule

A formula used to compute the derivative of a composite function, stating that if g is differentiable at x and f is differentiable at g(x), then the derivative of the composite function F(x) = f(g(x)) is F'(x) = f'(g(x)) * g'(x).

Theorem 1 of the Chain Rule

If g is differentiable at x and f is differentiable at g(x), then F(x) = f(g(x)) is differentiable at x and F'(x) = f'(g(x)) * g'(x).

Theorem 2 of the Chain Rule - Power Rule

If f(x) = [f(x)]^n, then f'(x) = nf(x)^(n-1)f'(x).

Exponential Function Derivative Rule

If f(x) = e^(f(x)), then f'(x) = f'(x)*e^(f(x)).

Logarithmic Function Derivative Rule

If f(x) = b^(f(x)), then f'(x) = f'(x) * ln(b) * b^(f(x)).

Trigonometric Derivative Rule for sine

If f(x) = sin(f(x)), then f'(x) = f'(x) * cos(f(x)).

Trigonometric Derivative Rule for cosine

If f(x) = cos(f(x)), then f'(x) = -f'(x) * sin(f(x)).

Trigonometric Derivative Rule for tangent

If f(x) = tan(f(x)), then f'(x) = f'(x) * sec^2(f(x)).

Tangent Line Equation

The equation of a tangent line at a point (x1, y1) is given as y - y1 = f'(x1)(x - x1), where f'(x1) is the derivative evaluated at x1.