Rules of Differentiation Quiz

Basic Derivative Rules:
- Power Rule: ( \frac{d}{dx}(x^n) = nx^{n-1} )
- Constant Rule: ( \frac{d}{dx}(c) = 0 ) where ( c ) is a constant.
- Constant Multiple Rule: ( \frac{d}{dx}(cf(x)) = c \cdot f'(x) )
Sum and Difference Rules:
- ( \frac{d}{dx}(f(x) + g(x)) = f'(x) + g'(x) )
- ( \frac{d}{dx}(f(x) - g(x)) = f'(x) - g'(x) )
Product Rule:
- ( \frac{d}{dx}(f(x)g(x)) = f'(x)g(x) + f(x)g'(x) )
Quotient Rule:
- ( \frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2} )
Chain Rule:
- ( \frac{d}{dx}(f(g(x))) = f'(g(x))g'(x) )

Finding Slopes: Derivatives provide the slope of the tangent line to a curve at a given point.
Optimization: Used to find maximum and minimum values of functions by setting ( f'(x) = 0 ).
Motion Analysis: Derivatives describe velocity and acceleration in physics.
Curve Sketching: Analyzing critical points and inflection points aids in sketching functions.

Definition: Technique to differentiate equations not explicitly solved for one variable in terms of another.
Process:
- Differentiate both sides of the equation with respect to ( x ).
- Treat ( y ) as a function of ( x ) (apply the chain rule).
- Solve for ( \frac{dy}{dx} ).

Definition: Derivatives of derivatives.
Notation:
- Second derivative: ( f''(x) = \frac{d^2}{dx^2}f(x) )
- Higher derivatives: ( f^{(n)}(x) ) for the ( n )-th derivative.
Applications:
- Assessing concavity and points of inflection with the second derivative test.
- Analyzing motion with acceleration being the second derivative of position.

Logarithmic Differentiation:
- Useful for products and quotients; take the natural log and differentiate.
Implicit Function Theorem:
- Helps to differentiate functions defined implicitly.
Parametric Differentiation:
- Derivatives of parametric equations ( x(t) ) and ( y(t) ) found using ( \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} ).
Numerical Differentiation:
- Approximating derivatives using difference quotients when the function is not easily differentiable.
Newton's Method:
- An iterative method for finding successively better approximations to the roots (zeroes) of a real-valued function, relying on derivatives.

Basic Derivative Rules include essential techniques for various functions:
- Power Rule: Enables differentiation of functions of the form ( x^n ).
- Constant Rule: Implies that the derivative of any constant is zero.
- Constant Multiple Rule: Allows for differentiation of a function scaled by a constant.
Sum and Difference Rules facilitate operations involving multiple functions:
- The derivative of the sum of two functions is the sum of their derivatives.
- The derivative of the difference of two functions is the difference of their derivatives.
Product Rule: A method to find the derivative of the product of two functions, requiring knowledge of both derivatives and the original functions.
Quotient Rule: Utilized for finding the derivative of a ratio of two functions, incorporating both derivatives and the original ratio.
Chain Rule: Essential for differentiating composite functions, relating the derivatives of the outer and inner functions.

Finding Slopes: Derivatives determine the slope of tangent lines at specific points on curves, essential in calculus.
Optimization: Used for identifying local maxima and minima by solving ( f'(x) = 0 ).
Motion Analysis: Derivation plays a critical role in kinematics, defining velocity (first derivative) and acceleration (second derivative).
Curve Sketching: Critical points and inflection points derived from differentiation enhance the visualization of function behavior.

Definition: A technique for differentiating equations where one variable is not isolated.
Process:
- Differentiate both sides with respect to ( x ) while treating other variables appropriately.
- Use the chain rule to express derivatives accordingly and isolate ( \frac{dy}{dx} ).

Definition: Refers to derivatives taken multiple times.
Notation:
- The second derivative of a function ( f ) is expressed as ( f''(x) ) and higher derivatives as ( f^{(n)}(x) ).
Applications:
- Analyze function concavity and identify inflection points with the second derivative test.
- Motion analysis: Understand acceleration as the second derivative of position.

Logarithmic Differentiation: Effective for complex products and quotients by logarithmic transformation prior to differentiation.
Implicit Function Theorem: Assists in differentiating implicitly defined functions, expanding the types of functions that can be analyzed.
Parametric Differentiation: For parametric equations, calculate derivatives using time parameterization to relate ( \frac{dy}{dx} ) through ( \frac{dy}{dt} ) and ( \frac{dx}{dt} ).
Numerical Differentiation: Approximates derivatives via difference quotients for functions that lack a clean differentiable form.
Newton's Method: An iterative numerical technique for approximating roots of a function, involving derivatives to refine guesses accurately.