5.1 DERIVATIVES OF EXPONENTIAL FUNCTIONS By: Rafal, Paola , Mujahid RECALL: y=ex (exponential function) LOGARITHM FUNCTION IS THE
INVERSE EX1: y=log4 x Therefore y=ex y=4x y=log e x IN THIS SECTION: The values of the derivative f(x) are
the same as those of the original function y=ex THE FUNCTION IS ITS OWN DERIVATIVE f(x)=ex f(x)=ex e is a constant called Eulers number or the natural number, where e= 2.718
The product, quotient, and chain rules can apply to exponential functions when solving for the derivative. f(x)=e g(x) Derivative of composite function: f(x)=e g(x) f(x)= e g(x) g(x) by using the chain rule Key Points f(x)= ex , f(x)= ex
Therefore, y= ex has a derivative equal to itself and is the only function that has this property. The inverse function of y=ln x is the exponential function defined by y= ex. Example 1: Find the derivative of the following functions.
a) y= e 3x+2 b) y= ex2+4x-1 y= g(x) (f(x)) y= g(x) (f(x)) y=(3)(e 3x+2 ) y= (2x+4) (ex2+4x-1 ) y= 3 e 3x+2 y= 2(x+2)(ex2+4x-1 ) You can also use the product rule and
quotient rule when appropriate to solve. Recall: Product rule: f(x)= p(x)(q(x)) + p(x)(q(x)) Quotient rule: f(x)=p(x)(q(x)) p(x)(q(x)) ___________________________ q(x)2 Example 2: Find the derivative and simplify a)
f(x)= X2e2x f(x)= 2(x)(e2x ) + (X2 ) (2)(e2x ) use product rule f(x)= 2x e2x + 2 X2 e2x simplify terms f(x)=2x(1+x) e2x factor out 2x b) f(x)= ex x
f(x)= (1) (ex )(x) (ex )(1) X2 use quotient rule f(x)= x ex - ex X2 simplify terms f(x)= (x-1) (ex ) factor out ex ______ X2 Example 3: Determine the equation of the line tangent to f(x) = exx2 , where x= 2. Solution: Use the derivative to determine the slope of the required tangent.
f(x) = exx2 f(x)=x-2ex Rewrite as a product f(x)= (-2x-3)ex + x-2 (1)ex Product rule f(x)= -2ex x3 + ex x2 Determine common denominator f(x)=-2ex x3 + xex x3 f(x)= -2ex + xex x3 f(x)= (-2 + x) ex x3 Simplify Factor When x=2, f(2) = e2 4. When x=2,
f(2)=0 . so the tangent is horizontal because f(2)=0. Therefore, the equation of the tangent is f(2) = e2 4 Example 4: The number, N, of bacteria in a culture at time t, in hours, is N(t)=1000(30+e-). a) b)
c) What is the initial number of bacteria in the culture? Determine the rate of change in the number of bacteria at time t. What is happening to the number of bacteria in the culture as time passes?
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