MAT.126
2.2-2.3

2.2 Basic Differentiation Rules and Rates of Change

Find the derivative of a function using the Constant Rule

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

Find the derivative of a function using the Power Rule

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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.

x

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

Rewriting

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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.

Evaluating the derivative to find the slope at a point

Finding the equation of a tangent line

Find the derivative of a function using the Constant Multiple Rule

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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 Sum and Difference Rules

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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 derivatives of the sine function and of the cosine function

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d/dx[sin x] = cos xd/dx[cos x] = -sin x

Use derivatives to find rates of change

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Average velocity = secant line (no limit)Instantaneous velocity = tangent line (limit)Speed is the absolute value of velocity (velocity is a vector quantity).

2.3 Product and Quotient Rules and Higher-Order Derivatives

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The derivative is important to three of the four major problems that led to the development of calculus.The tangent line problemThe velocity and acceleration problemThe minimum and maximum problem

Find the derivative of a function using the Product Rule

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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

Find the derivative of a function using the Quotient Rule

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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)<>0dx g(x) [g(x)]^2

Algebra!

Use lots of parentheses

Rewrite when necessary

Constant Multiple Rule

Simplify

Find the derivative of a trigonometric function

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d---[ tan x ] = sec^2 xdx d---[ cot x ] = -csc^2 xdx d---[ sec x ] = sec x tan xdx d---[ csc x ] = -csc x cot xdx

Trigonometric Identities

Find a higher-order derivative of a function

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Taking derivatives of derivatives.PositionVelocityAccelerationJerkWe'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.