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Rules of derivative
- 2. The process ofFinding average rate of change in the value of a variable with respective another variable is called Derivation. What is Differentiation/Derivative? Notation for Derivative Y = f (x)
- 3. Bye ab-initioor by definition We can find derivative of a given function by following Methods By using Direct method (by rules of derivative)
- 4. For a givenfunction f(x), f’(x) can be found by following four steps:- Derivative By Definition Step I Find f(x+&x) Step II Find f(x+&x)-f(x) Step III Find f(x+&x)-f(x) by &x to get f(x+&x)-f(x) and simplify it. &x Step IV lim f(x+&x)-f(x) &x 0 &x
- 5. Product Rule Rules ofDerivative Quotient Rule Power Rule Trigonometric Derivative Derivative of exponential functions Chain Rule
- 6. • Derivative of(xn) is found by: n ∙ xn-1 • Multiply (x) by the exponent (n) then subtract 1 from the original exponent (n). • Example: 1. if f(x) is x3 find f’(x). 3 ∙ x3-1 3x2 is the derivative. Power Rule
- 7. • To findthe derivative of a problem where we need to multiply (example: (x3) (x2+2)) then we use the product rule: y = u・v y’ = u・v’ + v・u’ y= (x3)(x2+2) u= x3 • y’= (x3)(2x)+ (x2+2)(3x2) v= (x2+2) u’= 3x2 v’=2x Product Rule
- 8. For trig functionswe do not use the power rule, instead they each have their own derivative. y y’ sin x cos x cos x -sin x tan x sec2x cot x -csc2x sec x sec x tan x csc x -csc x cot x Trigonometric Derivatives
- 9. We use chainrule when there is a problem usually inside a parenthesis (then raised to a power), or under a radical or in the denominator. We substitute this with the letter u. so we solve the derivative for u and later replace it for the original function. For example: y=(x3+1)9 u=x3+1 y=(u)9 du=3x2 dy=9u8 dx du Chain Rule
- 10. y= ex y’= ex∙ d x dx Example: y= e2x+1 y’= e2x+1 ∙ 2 y’= 2e2x+1 Derivative of Exp function
- 11.