类别 全部 - derivatives - tangent - differentiation - functions

作者:David Kedrowski 14 年以前

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MAT.126 2.2-2.3

Differentiation involves a set of rules for finding the derivatives of various types of functions, which are crucial in understanding rates of change. The basic differentiation rules cover the derivatives of sine and cosine functions, with specific formulas for each.

MAT.126 2.2-2.3

MAT.126 2.2-2.3

2.3 Product and Quotient Rules and Higher-Order Derivatives

The derivative is important to three of the four major problems that led to the development of calculus.

  • The tangent line problem
  • The velocity and acceleration problem
  • The minimum and maximum problem
  • Find a higher-order derivative of a function

    Taking derivatives of derivatives.

    Position

    Velocity

    Acceleration

    Jerk

    We've been finding first derivatives.

    The second derivative is the derivative of the first derivative.

    The fifth derivative is the derivative of the fourth derivative.

    Find the derivative of a trigonometric function

    d

    ---[ tan x ] = sec^2 x

    dx

    d

    ---[ cot x ] = -csc^2 x

    dx

    d

    ---[ sec x ] = sec x tan x

    dx

    d

    ---[ csc x ] = -csc x cot x

    dx

    Trigonometric Identities
    Algebra!
    Simplify
    Constant Multiple Rule
    Rewrite when necessary
    Use lots of parentheses
    Find the derivative of a function using the Quotient Rule

    The product f/g of two differentiable functions f and g is itself differentiable at all values of x for which g(x) is not zero. Moreover, the derivative of f/g is given by the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator.

    d f(x) g(x) f'(x) - f(x) g'(x)

    ---[ ---- ] = ------------------------, g(x)<>0

    dx g(x) [g(x)]^2

    Find the derivative of a function using the Product Rule

    The product of two differentiable functions f and g is itself differentiable. Moreover, the derivative of fg is the first function times the derivative of the second, plus the second function times the derivative of the first.

    d

    ---[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)

    dx

    2.2 Basic Differentiation Rules and Rates of Change

    Use derivatives to find rates of change

    Average velocity = secant line (no limit)

    Instantaneous velocity = tangent line (limit)

    Speed is the absolute value of velocity (velocity is a vector quantity).

    Find the derivatives of the sine function and of the cosine function

    d/dx[sin x] = cos x

    d/dx[cos x] = -sin x

    Find the derivative of a function using the Sum and Difference Rules

    The sum (or difference) of two differentiable functions f and g is itself differentiable. Moreover, the derivative of f+g (or f-g) is the sum (or difference) of the derivatives of f and g.

    Find the derivative of a function using the Constant Multiple Rule

    If f is a differentiable function and c is a real number, then cf is also differentiable and d/dx[cf(x)]=cf'(x).

    Using parentheses when differentiating
    Find the derivative of a function using the Power Rule

    The derivative of a power function x^n is nx^{n-1} for n a rational number.

    For f to be differentiable at x=0, n must be a number such that x^{n-1} is defined on an interval containing 0.

    Finding the equation of a tangent line
    Evaluating the derivative to find the slope at a point
    Rewriting

    It is very useful to rewrite radicals into rational exponent form and to write variables in the denominator of a fraction in negative exponent form.

    x

    The derivative of x is 1.

    This follows from the fact that the slope of the line y=x is 1.

    Find the derivative of a function using the Constant Rule

    The derivative of a constant function is 0.

    This means that the slope of a constant function is 0.