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Derivatives

Derivatives. What is a derivative? . Mathematically, it is the slope of the tangent line at a given pt. Scientifically, it is the instantaneous velocity of a particle along a line at time, t. Or the instantaneous rate of change of a fnc . at a pt. We write:.

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Derivatives

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  1. Derivatives

  2. What is a derivative? • Mathematically, it is the slope of the tangent line at a given pt. • Scientifically, it is the instantaneous velocity of a particle along a line at time, t. • Or the instantaneous rate of change of a fnc. at a pt.

  3. We write: Formal Definition of a Derivative: is called the derivative of fat a. “The derivative of f with respect to x is …” There are many ways to write the derivative of

  4. “the derivative of f with respect to x” “f prime x” or “y prime” “the derivative of y with respect to x” or “dee why dee ecks” “the derivative of f with respect to x” or “dee eff dee ecks” “the derivative of f of x” “dee dee ecks uv eff uv ecks” or

  5. Note: dydoes not mean d times y ! dx does not mean d times x !

  6. does not mean ! does not mean ! Note: (except when it is convenient to think of it as division.) (except when it is convenient to think of it as division.)

  7. does not mean times ! Note: (except when it is convenient to treat it that way.)

  8. A function is differentiable if it has a derivative everywhere in its domain. The limit must exist and it must be continuous and smooth. Functions on closed intervals must have one-sided derivatives defined at the end points. p

  9. The derivative is the slope of the original function. The derivative is defined at the end points of a function on a closed interval.

  10. DIFFERENTIATION RULES: • If f(x) = c, where c is a constant, then f’(x) = 0 • If f(x) = c*g(x), then f’(x) = c*g’(x) • If f(x) = xn, then f’(x) = nxn-1 • SUM RULE: • If f(x) = g(x) + h(x), then f’(x) = g’(x) + h’(x) • DIFFERENCE RULE: • If f(x) = g(x) - h(x), then f’(x) = g’(x) - h’(x) • PRODUCT RULE: • If f(x) = g(x) * h(x), then f’(x) = g’(x)*h(x) + h’(x)*g(x) • QUOTIENT RULE: • f(x) = then f’(x) = • CHAIN RULE: • If f(x) = g(h(x)), then f’(x) = g’(x)*h’(x)

  11. Derivatives to memorize: • If f(x) = sin x, then f’(x) = cos x • If f(x) = cos x, then f’(x) = -sin x • If f(x) = tan x, then f’(x) = sec2x • If f(x) = cot x, then f’(x) = -csc2x • If f(x) = sec x, then f’(x) = secxtanx • If f(x) = csc x, then f’(x) = -cscxcotx • If f(x) = ex, then f’(x) = ex • If f(x) = ln x, then f’(x) = 1/x • If f(x) = ax, then f’(x) = (ln a) * ax

  12. Examples: Find f’(x) if • f(x) = 5 • f(x) = x2 – 5 • f(x) = 6x3+5x2+9x+3 • f(x) = (3x+4)(2x2-3x+5) • f(x) = • f(x) = • f(x) = (3x2+5x-2)8 • f(x)=

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