The Derivative of the Natural Logarithmic Function
Ex 1 Differentiate each function.Guess the derivative: Actual derivative:fx=Sin(2x)f(x)=3x+11y=5x-12 y=ex2- need new rule hereTheorem The Chain RuleIf fu is differentiable at the point u=g(x) and g(x) is differentiable at the point x, then the composite function f°gx=f(gx) is differentiable at x andf°g'x=f'gxg'(x).In Leibniz’s notation, if y=f(u) and u=g(x), thendydx=dydu?dudx , where dydu is evaluated at u=g(x).Note: “Outside-Inside” Rule is helpful way to remember Chain Rule.If y=fgx=f(u), then dydx=f'gx?g'x=f'(u)?u'. Derivative of outside Derivative of InsideThe “outside” function is f and the “inside” function is u=g(x). To take the derivative of the composite function, fgx, we:“ take the derivative of the outside times the derivative of the inside.”Note: When we differentiate the “outside” function f, we leave g(x) alone and plug it into f'.You can think of this as a unit conversion as seen in Leibniz’s notation: dydx=dydu?dudxUsing the Chain Rule, we see the following:If u is a function of x, then ddxeu=eududx.The General Power Ruleddxun=nun-1dudx, where u is a function of x.Note: Regarding the Chain Rule, practice makes perfect. Important to be solid with algebra.Ex 2 (# 4) Given y=f(u) and u=g(x), where f(u)=cosu, u=-x3 find 1) find fu 2) Find dydxEx 3 Write the function in the form y=f(u) and u=g(x). Then find dydx as a function of x.(# 14) y=x5+15x5(# 22) y=e4x+x2Ex 4 Find the derivative of each function. Find y'' for part c).(#49) y=Sin2(πt-2) (# 60) y=4sin1+t (# 38) y=9x2-6x+2ex3(# 66) y=sin(x2ex)Ex 5 (# 70) Find the value of f°g' at the given value of x. fu=u+1cos2u, u=gx=πx, x=14 The Derivative of the Natural Logarithmic FunctionTheorem Derivative of the Natural logarithmic functionddxlnx=1x for x>0 ddxlnx=1x or ddxlnu=1ududx=u'u, u>0 In particular, ddxlnx=1x?1 ddxlnu=1ududx=u'u Proof:Ex 5 Find the derivative of y wrt x, t, or θ, as appropriate.y=ln10xy=lnkx, k constant y=ln(t3/2) y=lnx3y=t2ln3t y=ln1+lnlnxy=lnsinθcosθ1+2lnθ (Recall: lnxy=lnx+ln y and lnxy=lnx-lny & Sin 2x=2Sinx Cosx)Bases Other than eDefinition of Exponential function to Base aIf a is a positive real number (a≠1) and x is any real number, then the exponential function to the base a is denoted by ax and is defined byax=elnaxIf a=1, then y=1x is the constant function y=1.Definition of Logarithmic Function to Base aIf a is a positive real number (a≠1) and x is any real number, then the logarithmic function to the base a is denoted by logax and is defined bylogax=lnxlnaTheorem Derivatives for Bases other than eddxau=aulnadudx In particular, ddxax=axlnaddxlogau=1lna?1ududx In particular, ddxlogax=1xlnaProof: First proof Uses the fact that limh→0ah-1h=lna Ex 6 Find the derivative wrt to the independent variable.(# 78) y=log3r?log9r (Use Change of Base – change to base e)(# 86) y=3log8log2t (Use Change of Base – change to base e)y=73x A Brief Discussion on Parametric Equations (Book does not cover this section until later)Given a curve in 2-space as in Figure 3.29Describing this curve can be difficult, especially if the curve cannot be described by a 50057055715Function. To remedy this, we can use Parametric Equations to describe the curve.Def: Parametric CurveIf x and y are given as functions x=ft & y=g(t)Over an interval of t-values (i.e. on domain of t) then any point (x,y) in 2-D can bedefined by the equivalent point (ft,gt) and in this format the curve is a parametric curve. The equations are Parametric Equations for the curve.A physical example of parametric equations is and etch-a-sketch.Think of how an etch-a-sketch works. One knob is the parametric equation for the x direction, the other knob is the parametric equation for the y direction. To move the sketcher you must independently rotate (t) the knobs.Then the position of the sketch is a function of the x-knob and the y-knob.Ex: Mathematically describe a circle centered at the origin with radius=1 using only one variable.What we need to do is to use a single variable to describe every point on the circle. We will use the hour hand of a clock moving counter clock wise as the idea. The variable will be the angle of the hour hand.From Trig recall that any point on a unit circle can be described by the angle the radius makes with the positive x axis. (rcosθ,rsinθ) This is the general idea. So.The curve described by x2+y2=1Can be described by the parametric equations, x=1?Cosθ and y=1Sinθ for 0≤θ≤2πTo check we plug in our parametric equations into the equation:x2+y2=Cosθ2+Sinθ2=1Ex: Mathematically describe an ellipse centered at the origin with vertical radius of 2 and horizontal radius of 3.I.E. Describe the curvex29+y24=1By using parametric equationsLet x=3Cosθ and y=2Sinθ for 0≤θ≤2πThenx29+y24=3Cosθ29 +(2Sinθ)24=9Cos2θ9 +4Sin2θ4=Cos2θ+Sin2θ=1Ex: The position p(x,y) of a particle moving in the xy-plane is given by the equations and parameter interval x=t;y=t for t≥0Identify the path traced by the particle and describe the motion.y=t=t2=x2 So the parametric equations are describing the right half of a parabola w/ vertex at (0,0)Derivatives of Parametric Curves:A parameterized curve, x=ft & y=g(t) is differentiable at t if f & g are differentiable at tDerivatives of Parametric Curves:A parameterized curve, x=ft & y=g(t) is differentiable at t if f & g are differentiable at t, and is given by: dydt=dydx?dxdt→Provided that all three derivatives exist and dxdt≠0.dydx=dydtdxdtParametric Formula for d2ydx2:If all the equations x=ftand y=gtdefine y as twice-differentiable function of x, then at any point where dxdt≠0,then d2ydx2=dy'dtdxdt ................
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