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Differentiation Safdar Alam Table Of Contents • • • • • Chain Rules Product/Quotient Rules Trig Implicit Logarithmic/Exponential Notations of Differentiation • In functions you will see: -f’(x) -y’(x) • These symbols are used to tell that the function is a derivative • Derivative: lim h> 0 f(x+h)–f(x) h Formula for Derivative • Nu↑(n-1) • N, standing for a constant (which a derivative of a constant is zero) • U, standing for a function • Example: X₂ • Answer: 2x Practice Problems • F(x)= 5x₄ • F’(x)= • F(x)= x₂+3x₂ • F’(x)= Work Page Definition: Formula for the derivative of the two function There are two types of chain rules. (Product/Quotient Rule) Product: (F*DS + S*DF) Quotient: (B*DT – T*DB) B₂ Chain Rules Product Rule • Used for Multiplication • Product: (F*DS + S*DF) • (First * Derivative of Second + Second * Derivative of First) • • • • • Example: Y = (4x + 3)(5x) Y’= (4x + 3)(5) + (5x)(4) Y’= (20x + 15) + (20x) Y’= 40x + 15 Practice Problem • Y= (6x + 4)₂(25x + 13) • Y’= Work Page Used for Division Quotient: (B*DT – T*DB) B₂ (Bottom * Derivative of Top – Top * Derivative of Bottom over Bottom Squared) Example F(x) = (5x + 1) x F’(x) = (x)(5) – (5x + 1)(1) – (5x + 1) x₂ x₂ F’(x) = ( -1 ) x₂ Quotient Rule F’(x) = (5x) Practice Problem • F(x)= • F’(x)= 2 , (4x + 1)₂ Work Page Derivative of Trig. Functions Sin(x) = Cos(x) dx Cos(x) = -Sin(x) dx Sec(x)= (secx)(tanx) dx Tan(x)= Sec₂(x) dx Csc(x)= -(cscx)(cotx) dx Cot(x)= -csc₂(x) dx Example: Y= cos(x) + sin(x) Y’= -sinx + cosx Trig Functions Practice Problems • Y= tanx sinx • Y’= Work Page We use implicit, when we can’t solve explicitly for y in terms of x. Example: Y ₂ = 2y dy dx Implicit Differentiation Practice Problems • F(x) = x₃ + y₃ = 15 • F’(x) = Work Page This applies to chain rules and properties of logs Rules of Log Multiplication- Addition Division- Subtraction Exponents- Multiplication Some key functions to remember ln(1) = 0 ln(e) = 1 ln(x)x = xln(x) Logarithmic Differentiation Practice Problems • Y= (3)x • Dy/Dx= Work Page F’(x) e(u)= e(u) (du/dx) -Copy the Function and take the derivative of the angle Examples: Y = e(5x) Y’= 5e(5x) Y= e(sinx) Y’= e(sinx)*(-cosx) Exponential Diff. Practice Problems • F(x)= e(5x + 1) • F’(x) = • F(x)= e(tanx) • F’(x)= Work Page Y= ln(x) Y’(x)= 1/u * du/dx Examples Y= ln(5X) Y’= 5/5X = 1/X Derivative of Natural Log Practice Problems • Y= ln(ex) • Y’= • Y=ln(tanx) Work Page FRQ • 1995 AB 3 -8x₂ + 5xy + y₃ = -149 A. Find dy/dx -16x + 5x(dy/dx) + 5y + 3y₂(dy/dx) = 0 (dy/dx)(5x + 3y₂) = 16x – 5y dy/dx = 16x – 5y 5x + 3y₂ FRQ • 1971 AB 1 - ln(x₂ - 4) E. Find H’(7) 1 * (2x) (X₂ - 4) 2(7) ( (7)₂ - 4 ) 2x . (X₂ - 4) 14 . (49 – 4 ) 14 45 Sources • • • http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/implicitdiffdirectory/ImplicitDiff.html http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/logdiffdirectory/LogDiff.html http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/chainruledirectory/ChainRule.html © Safdar Alam March 4, 2011