What's More Solve for the derivatives of the following functions with complete solutions. Use a separate sheet of paper for your answer. 1. y=1n(5x2) 2. y=1n(45x) 3. y=e\""3" 4. y=3-"z 5. y=1nx= y = In (5x) EXAMPLE 2. Find the derivative of y = In (x3 + 4). Solution: y = In (x3 +4) dy 1 d dx x3 +4 dx (* + 4) as = 1. (3x2) dx x3+4 dy 3x2 dx x3 +4 y = In (x3 + 4)Differentiating a Logarithmic Function Expressions written in exponential form can be converted to logarithmic function and vice versa. Exponential Form to Logarithmic Form 53 = 125 log5 125 = 3 490.5 # 7 = log49 7 = 0.5 Logarithmic Form to Exponential Form log 2 8 = 3 23 = 8 log3 81 = 4 34= 81 Hence y = log, x can be written as by = x and y = log, x can be written as ey = x. natural logarithms are to the base e In x is used for natural logarithms The derivative of the Natural Logarithm Function If y = In x, then ~ Inx = = If u is a differentiable function of x, then according to the Chain Rule: dx " Inu = - Inu.- d du du dx -Inu = =. 1 du dx u dx Derivative of Logarithmic Functions other than the natural logarithms day (logb x) = 1 x In b If u is a differentiable function of x, then d 1 du dx - (logb u) = u Inb dx EXAMPLE 1. Find the derivative of y = In (5x). Solution: Use A Inu = 1. au y = In (5x) 1 d 5x dy (5x) dy 5 dx 5x orEXAMPLE 1. Now let us go back at our illustration in What's New. We can now solve for the slope of the tangent line by finding the derivative of y = x3 + 3* using the differentiation rule above. Solution: dax = a* Ina dx Formula (x3 + 3*) = 3x2 + 3* (In 3) Solved the derivative of x3 and 3x. EXAMPLE 2. Find the derivative of y = ex. Solution: Let u = ex Cell = qu du dx dx Formula dy _ = pex da = eex d (ex) Differentiation is linear. Differentiated them separately and pulled out constant factors = eex . e . " (x) Derivative of x is 1 = gex+1 Final answer EXAMPLE 3. Differentiate y = x . Inx - x Solution: dx " = x . In x - x = " (xInx) - -(x) Differentiation is linear. Differentiated them separately and pulled out constant factors = (x) . Inx + x . -(Inx) - 1 Applied the product rule = 1Inx+x .--1 Simplified (derivative of In x is 1/x) = In x Final answer EXAMPLE 4. Find the derivative of y = e 4x+7. Solution: dx - ell = eu du Formula dx =ex+7_M dx - (4x + 7) == e4x+7 4 . 4 (x) + + (7)] Differentiation is linear. Differentiated them separately and pulled out constant factor = ex+7(4 . 1+0) Differentiated each term = 4e4x+7 Final answeriiirrmmmrms AN mommm runcriou ns that we like to focus on are eiponential and logarithmic 'l'he next 'set of functio mmonly need exponential form in a calculus course is the functions. The most (:0 natural exponential function 9". 0n the other hand, the process of differentiating functions by taking logarithms rst and then differentiating is called logarithmic differentiation. We utilize logarithmic differentiation in circumstances where it is easier to differentiate the logarithm of a function than to differentiate the function itself. This approach \"allows calculating derivatives of power, rational and Some irrational functions in an efcient manner. We will start off by looking at the exponential function. For the natural exponential function, f (x) = ex we have n e 1 I ' __- = f (0) ' 93% h 1 Provided we are using the natural exponential function, we get the following: rm = ex = [\"00 = e Now, we are missing some skills that will permit us to simply get the derivative for a general function. Eventually, we will be able to show that for a general exponential function, we will have: f(x) = a' =? f '(x) = Wu (6!) *\"The derivative of the exponential function f (x) = y = eJr is its own function. This can be linked with the chain rule. If u is a function of x, then (1 du _ u = u_ dxe 8 dx' Summary of Derivative of Exponential Functions :1 ._ 1.. x d e ' a" =axlna M 11