differentiation safdar alam. table of contents chain rules product/quotient rules trig implicit...
TRANSCRIPT
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 f ( x + h ) – f ( x ) h> 0 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
Chain Rules
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
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
Quotient Rule
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) F’(x) = (5x) – (5x
+ 1) x₂ x₂ F’(x) = ( -1 ) x₂
Trig Functions
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
Implicit Differentiation
We use implicit, when we can’t solve explicitly for y in terms of x.
Example:
Y ₂ = 2y dy dx
Logarithmic Differentiation
This applies to chain rules and properties of logs Rules of Log
Multiplication- AdditionDivision- SubtractionExponents- Multiplication
Some key functions to rememberln(1) = 0ln(e) = 1ln(x)x = xln(x)
Exponential Diff.
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)
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 – 5ydy/dx = 16x – 5y
5x + 3y₂
FRQ
• 1971 AB 1 - ln(x₂ - 4)
E. Find H’(7) 1 * (2x) 2x . (X₂ - 4) (X₂ - 4) 2(7) 14 . 14 ( (7)₂ - 4 ) (49 – 4 ) 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