Sheng-Fang Huang

Definition of Derivative The Basic Concept

Definition of Derivative Definition Let y = f(x) be a function. The derivative of f is the function whose value at x is the limit: provided this limit exists. The derivative of f is also written as f ’, or df/dx. If this limit exists for each x in an open interval I (e.q. from -∞ to + ∞), then we say that f is differentiable on I.

Standard Derivative

Exercises Compute dy/dx

Functions of a function (1) sinx is a function of x since the value of sinx depends on the value of the angle x. Simiarly, sin(2x+5) is a function depends on ________. Since (2x+5) is also a function of x, we say sin(2x+5) is a function of a function of x. By Chain Rule: Let y = sin(2x+5), u = 2x+5.

Functions of a function (2) Products Compute the derivative of e 2x ln5x. Quotients Compute the derivative of.

Logarithmic Differentiation The rule for differentiating a product or a quotient is used when there are only two factors, i.e. uv or u/v. Where there are more than two functions, the derivative is best found by ‘logarithmic differentiation’. Let, where u, v, and w are functions of x. Take logs to the base e on both sides:

Exercises Solve the derivative of. y = x 4 e 3x tanx.

Implicit Functions Explicit function: If y is completely defined in terms of x, y is called an explicit. E.g. y = x 2 – 4x + 2 Implicit function E.g. x 2 + y 2 = 25, or x 2 + 2xy + 3y 2 = 4. The differentiation of an implicit function:

Partial Differentiation The volume V of a cylinder of radius r and height h is given by V depends on two quantities, r and h. Keep r constant, V increases as h increases. Keep h constant, V increases as r increases. is called the partial derivative of V with respect to h where r is constant. Let z = 3x 2 + 4xy – 5y 2. Compute and.

Standard Integrals (1)

Function of a linear function of x We know that, but how about ? Solution: Let z = 5x-4. Change the original equation into the following form: We have Now, try to solve and.

and Formula: How to prove? Exercises:

Integration of products How to integrate x 2 lnx? Integration by parts: Solve.

Integration by partial fractions Example: Rules: degree of numerator < degree of denominator If not, first of all, divide out by the denominator. Denominator must be factorized into prime factors (important!). (ax+b) gives partial fractions A/(ax+b) (ax+b) 2 gives partial fractions A/(ax+b) + B/(ax+b) 2 (ax+b) 3 gives partial fractions A/(ax+b) + B/(ax+b) 2 + C/(ax+b) 3 ax 2 +bx+c gives partial fractions (Ax+B)/(ax 2 +bx+c)

Integration by partial fractions Example:

Integration of Trigonometrical Functions Basic trigonometrical formula:

Integration of Trigonometrical Functions

Multiple Integration Double integral A double integral can be evaluated from the inside outward. Key point: when integrating with respect to x, we consider y as constant. 1 2

Multiple Integration A double integral can also be sometimes expressed as:

Multiple Integration Example If. Compute V.