Compute the derivative by definition: The four step procedure


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1 Compute te derivative by definition: Te four step procedure Given a function f(x), te definition of f (x), te derivative of f(x), is lim 0 f(x + ) f(x), provided te limit exists Te derivative function f (x) is sometimes also called a slopepredictor function Te following is a fourstep process to compute f (x) by definition Input: a function f(x) Step Write f(x + ) and f(x) Step Compute f(x + ) f(x) Combine like terms If is a common factor of te terms, factor te expression by removing te common factor f(x + ) f(x) Step 3 Simply As 0 in te last step, we must cancel te zero factor in te denominator in Step 3 f(x + ) f(x) Step 4 Compute lim by letting 0 in te simplified expression 0 Example Let f(x) ax + bx + c Compute f (x) by te definition (tat is, use te four step process) Solution: Step, write f(x+) a(x+) +b(x+)+c a(x +x+ )+bx+b+c ax +ax+a +bx+b+c Step : Use algebra to single out te factor f(x+) f(x) (ax +ax+a +bx+b+c) (ax +bx+c) ax+a +b (ax+a+b) Step 3: Cancel te zero factor is te most important ting in tis step f(x + ) f(x) (ax + a + b) Step 4: Let 0 in te resulted expression in Step 3 ax + a + b f f(x + ) f(x) (x) lim lim ax + a + b ax b a + b 0 0 Example Let f(x) x + Compute f (x) by te definition (tat is, use te four step process)
2 Solution: Step, write f(x + ) Step : Use algebra to single out te factor f(x + ) f(x) (x + ) + x + + x + + (x + ) (x + + ) x + (x + + )(x + ) Step 3: Cancel te zero factor is te most important in tis step f(x + ) f(x) [ x + + [ x + (x + + )(x + ) Step 4: Let 0 in te resulted expression in Step 3 (x + + )(x + ) (x + + )(x + ) f f(x + ) f(x) (x) lim lim 0 0 (x + + )(x + ) (x )(x + ) (x + ) Example 3 Let f(x) x + 5 Compute f (x) by te definition (tat is, use te four step process) Solution: Step, write f(x + ) (x + ) + 5 x Step : Use algebra to single out te factor Here we need te identity (A + B)(A B) A B to get rid of te square root so tat can be factored out f(x+) f(x) x x + 5 (x + + 5) (x + 5) x x + 5 Step 3: Cancel te zero factor is te most important ting in tis step f(x + ) f(x) [ x x + 5 Step 4: Let 0 in te resulted expression in Step 3 f f(x + ) f(x) (x) lim lim 0 0 x x + 5 x x + 5 x x + 5 x x + 5 x + 5
3 Find an equation of te tangent line (using te 4step procedure to find slopes) Given a curve y f(x) and a point (x 0, y 0 ) on it, an equation of te line tangent to te curve y f(x) at te point (x 0, y 0 ) is y y 0 f (x 0 )(x x 0 ), provided te f (x 0 ) exists (Terefore, f (x 0 ) is te slope of te tangent line at (x 0, y 0 )) Example Let f(x) 4x + 5x + 6 Find an equation of te line tangent to te curve y f(x) at (, 5) Compute f () by te definition (tat is, use te four step process) Solution: Step, write f( + ) 4( + ) + 5( + ) + 6 4( + + ) Step : Use algebra to single out te factor f( + ) f() ( ) (3 + 4) Step 3: Cancel te zero factor is te most important ting in tis step f(x + ) f(x) (3 + 4) Step 4: Let 0 in te resulted expression in Step 3 Terefore, te answer is f f(x + ) f(x) (x) lim lim y 5 3(x ) Example Let f(x) Find an equation of te line tangent to te curve y f(x) x + at te point were x Compute f () by te definition (tat is, use te four step process) Solution: Step, write f( + ) ( + )
4 Step : Use algebra to single out te factor f( + ) f() + + ( + ) ( + + ) + ( + + )( + ) (3 + )(3) Step 3: Cancel te zero factor is te most important in tis step f( + ) f() [ 3 + [ 3 (3 + )(3) (3 + )(3) Step 4: Let 0 in te resulted expression in Step 3 f f(x + ) f(x) (x) lim lim 0 0 (3 + )(3) (3 + 0)(3) 9 Terefore, te slope m 9 As y 0 f() 3, te answer is y 3 (x ) 9 Example 3 Let f(x) x + 5 Find an equation of te line tangent to te curve y f(x) at te point were x Compute f () by te definition (tat is, use te four step process) Solution: Step, write f( + ) ( + ) Step : Use algebra to single out te factor Here we need te identity (A + B)(A B) A B to get rid of te square root so tat can be factored out f(x + ) f(x) (9 + ) Step 3: Cancel te zero factor is te most important ting in tis step f(x + ) f(x) [ Step 4: Let 0 in te resulted expression in Step 3 f f(x + ) f(x) (x) lim lim Terefore, te slope m 3 As y 0 f() 3, te answer is y 3 (x )
5 Example 4: Let f(x) be given x (a) Use definition of te derivative to find f (x) (b) Find an equation of te line tangent to te curve y f(x) at te point were x 3 Solution: (a) Step : Compute f(x + ) (x + ) x + Step : Compute te difference f(x + ) f(x) (We must ave as a common factor in te result) f(x + ) f(x) x + x x (x )(x + ) x + (x )(x + ) (x ) (x + ) (x )(x + ) (x )(x + ) Step 3: Use te result in Step to form and simplify te ratio (te denomination must be cancelled wit te numerator ) f(x + ) f(x) (x )(x + ) (x )(x + ) Step 4: Find te answer by letting 0: f f(x + ) f(x) (x) lim lim 0 0 (x )(x + ) (x )(x + 0 ) (x ) (b) First compute f(3) /(3 ) At te point (3, ), te slope of tangent line is f (3) ( )/(3 ) / Terefore an equation of te tangent line is Example 5: Let f(x) x + be given y (x 3) (a) Use definition of te derivative to find f (x) (b) Find an equation of te line tangent to te curve y f(x) at te point were x Solution: (a) Step : Compute f(x + ) (x + ) + x + + 5
6 Step : Compute te difference f(x + ) f(x) (We must ave as a common factor in te result) We sall utilizes te formula (A + B)(A B) A B (wic, as we ave seen, is a useful tool to deal wit square roots) f(x + ) f(x) x + + x + x + x + + x + x + + x + + x + x + x + + x + x + + x + + x + + x + + x + x + + x + x + + x + + (x + ) (x + + ) x + + x + ( x + + x + + ) x + + x + ( x + + x + + ) Step 3: Use te result in Step to form and simplify te ratio (te denomination must be cancelled wit te numerator ) f(x + ) f(x) x + + x + ( x + + x + + ) x + + x + ( x + + x + + ) Step 4: Find te answer by letting 0: f (x) lim 0 f(x + ) f(x) lim 0 (b) First compute f( ) x + + x + ( x + + x + + ) x x + ( x + + x ) (x + ) x + ( ) + At te point (, ), te slope of tangent line is f ( ) ( + ) + Terefore an equation of te tangent line is y (x ( )) 6
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