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Differential Calculus

# Differential Calculus - Differential Calculus Definition(x...

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Differential Calculus Definition ƒ ' ( x ) = and if this limit exists ƒ ' ( c ) = If ƒ is differentiable at x = c , then ƒ is continuous at x = c . Differentiation Rules General and Logarithmic Differentiation Rules 1. [ cu ] = cu' 2. [ u v ] = u' v' sum rule 3. [ uv ] = uv' + vu' product rule 4. [ ] = quotient rule 5. [ c ] = 0 6. [ un ] = nun -1 u' power rule 7. [ x ] = 1 8. [ln u ] = 9. [ eu ] = euu' 10. [ ƒ (g( x ))] = ƒ ' ( g ( x )) g' ( x ) chain rule Derivatives of the Trigonometric Functions 1. [sin u ] = (cos u ) u' 2. [csc u ] = -(csc u cot u ) u' 3. [cos u ] = -(sin u ) u' 4. [sec u ] = (sec u tan u ) u' 5. [tan u ] = (sec2 u ) u' 6. [cot u ] = -(csc2 u ) u' Derivatives of the Inverse Trigonometric Functions 1. [arcsin u ] = 2. [arccsc u ] =

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3. [arccos u ] = 4. [arcsec u ] = 5. [arctan u ] = 6. [arccot u ] = Implicit Differentiation Implicit differentiation is useful in cases in which you cannot easily solve for y as a function of x . Exercise : Find for y 3 + xy - 2 y - x 2 = -2 [ y 3 + xy - 2 y - x 2] = [-2] 3 y 2 + ( x + y ) - 2 - 2 x = 0 (3 y 2 + x - 2) = 2 x - y = Higher Order Derivatives These are successive derivatives of ƒ ( x ). Using prime notation, the second derivative of ƒ ( x ), ƒ '' ( x ), is the derivative of ƒ ' ( x ). The numerical notation for higher order derivatives is represented by: ƒ ( n )( x ) = y ( n ) The second derivative is also indicated by .
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