Chain Rule

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Chain Rule

« Product and Quotient Rule for Derivatives

Differentiation: (lesson 3 of 3)

If $y = f(u)$ is a differentiable function of $u$ , and $u = g(x)$ is a differentiable function of $x$ , then $y = f(g(x))$ . This a differentiable function of $x$ , and

\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}

or, equivalently,

\frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x)

Example 1:

\begin{aligned} y &= (x^2 + 1)^3, \ \frac{dy}{dx} = ? \\ g(x) &= u = x^2 + 1, \\ y &= f(u) = u^3, \\ \frac{df}{dx} &= \frac{df}{du} \cdot \frac{du}{dx} = (3u^2) (2x) = 3 (x^2 + 1)^2 (2x) = 6x (x^2 + 1)^2 \end{aligned}

The General Power Rule

\frac{d}{dx} [u(x)]^n = n [u(x)]^{n-1} \cdot u' (x)

Algebra

Roots and Radicals
Simplify Expression Adding and Subtracting Multiplying and Dividing
Complex Numbers
Arithmetic Polar representation
Polynomials
Multiplying Polynomials Division of Polynomials Zeros of Polynomials
Rational Expressions
Simplifying Multiplying and Dividing Adding and Subtracting
Solving Equations
Linear Equations Absolute Value Equations Quadratic Equation Equations with Radicals
Systems of Equations
Substitution Method Elimination Method Row Reduction Cramers Rule Inverse Matrix Method
Exponential Functions
Introduction Exponential Equations Logarithmic Functions
Trigonometry
Trigonometric Formulas Trigonometric Equations Law of Cosines
Progressions
Arithmetic Progressions Geometric Progressions

Calculus

Differentiation
Common formulas Product and Quotient Rule Chain Rule
Limits
Properties of Limits Rational Function Irrational Functions Trigonometric Functions L'Hospital's Rule
Integrals
Integration Formulas Exercises
Integral techniques
Substitution Integration by Parts Integrals with Trig. Functions Trigonometric Substitutions
Integral Applications
Area Volume Arc Length

Analytic geometry

Analytic Geometry 2D
Basic Concepts Lines Parallel and Perpendicular Lines Polar Coordinates
Conic Sections
Circle Ellipse Hyperbola
Analytic Geometry 3D
Line in 3D Planes

Linear Algebra

Matrices
Definitions Addition and Multiplication Gauss-Jordan elimination
Determinants
Introduction to Determinants Applications of Determinants
Vectors
Basic Operations Dot Product Cross Product

Random Quote

The essence of mathematics is its freedom.

Geoge Cantor

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Black holes result from God dividing the universe by zero.

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