Differentiation

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y₂ + ... + Yn-1) Az half strip width x [ends + twice the Y₁-1)] the middles] Querestimate s Turcare crestimate Negative and Fractional Powers Eg y = √x - 1 1/2 -x-¹ y = x dy da = 40 dx² d²y =o 2x² dy dx = x² Second Order differentiation d'y > o dz² -1/2 2√x = = + Differentiation Continued. +x minimum maximum y=uv Sketching the graphs of gradient functions" Chain Rule) dy = dy x dy du -2 dy x du dx du Product Rula It could be a maximum, minimum or a pout of inflection. - use the gradient method instead to identify stationary points. ←Write using powers before you differentiale. Eg. y=(x²+3)³ u=x^²+³ y=u³ 11 - use when there is a function inside a function. (often used in rates of change problems). Let u = the bit inside the function. dy dac Alternative method for identifying turning Concave upwards" Normal rules apply. Concave Downwarcis" 3u²x 2x dy = 34² du 6x4² 6x(x²+3)² d2 dy=2x = Eg. y = x² ex u= x² v=e dy = 2x dv=e* dx de -Need to know how to do this Quick way 3 y = (x²+3) ³ dy Idx = 3 X 2x X 4 7 power to the front vdu + udv da dx - use when two functions are multiplied together differential of brachat dy = 2xe* + x²e² dx = xẻ (2+x) xe Take I off the 2 power. (x²+3) ² Quotient Rule) y= u V Eg. y=x²+1 3x-1. dy = 2x(3x-1)= 3(x²+1) dz xponential y=e standard Results 4 must be on the top. must be on the bottom. 4=x²+1 du = 2x (3x-1)² No need to multiply it out. If you are looking for a stationary paint make the top = 0. Differentiating an Inverse Function y=ea* dy dx y = f(x) dy doc Egy=e*² dy da se product . على %/ay =x v=y - use when two functions form a =1 dv - dy dx dz : xdy ty da = ge Trig Functions y=sinx ax y = Sinax dy = a cosax dz - you need to learn! f'(x)e f(x) = 2xe²^² ax 83 da ay The gradient of f'(x) at (g;b) is | V = 3x -1. dv = 3. वर +y+³y²dy = dy=vdu - udv dx 2(x+3y²)=-y -५ x+3y². == 6x²-2x-3x²-3 (3x-1)²2 gradient of f(x) at (b, a). must be in radians. 4 y = cos x dy=-sinx y = cosax dy=-asinax y. Implicit Differentiation - Differentiating with 2 letters, usually scand #Be on the look out for the produce or quotient redes, when differentiating with the wrong letter, "superglua" a dy by the side of the gradient at (1,1) and the egn. of =g. xy + y² = 2. Find dy, the bangent. d =O dy = 0, you only need to › dx fraction. Natural Loganthme y=lnx = 1 y=in as dac du = 1/1/201 = 3x²-2x-3 (3x+1)² y=in|f(x)| 2. Egy = |n|x²+1 = 220 +1 Ho Be couefuul. dy = f'(x) x+ 4y = 5. F(x). y=tanx. dy कट y-1 = = =(x-1) y=1=-=1/3x²+4/5 y=-= x + 1 + = sec¹² I. I