12.2 Derivatives of Exponential Functions

Objective: To develop a differentiation formula for y= e^u, to apply the formula, and to use it to differentiate an exponential function with a base other than e

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EX 1: Differentiating Functions involving e^x 1) y= 3e^x 2) y= x/e^x 3) y= e^2 + e^x + ln3

1) = 3e^x 2) = 1 - x / e^x *we could have first used the quotient rule and then equation 2, but it is easier first to rewrite the function as y= xe^-1 and then use the product rule and equation 3 (e^u= e^u * du/dx) 3) = e^x (e^2 and ln3 are constants so they =o) y'= 0 + e^x + 0 = e^x

EX 4: Differentiating an Exponential Funct. with base 4 1) y= 4^x

1) = 4^x(ln4)

Ex 2: Differentiating Functions Involving e^u 1) y= e^(x^3 + 3x) 2) y= e^(x+1) * ln(x^2+1)

1) the function has the form e^u with u= x^3 + 3x. from equation 2, = 3(x^2+1)e^(x^3+3x) 2) by product rule, = e^(x+1) * (2x/x^2+1 +ln(x^2+1))

Combining Rule 6: Inverse Function Rule

If f is an invertible, differentiable function, then f^-1 is differentiable and, d/dx(f^-1(x))= 1/f '(f^-1(x))

Differentiating Exponential Functions to the Base b

Now that we are familiar with the derivative of e^u, we consider the derivative of the more general exponential function b^u. Because b= e^lnb, we can express b^u as an exponential function with the base e, a form we can differentiate. Equation 4: d/dx(b^u)= b^u * (lnb) * du/dx Procedure to differentiate b^u: Convert b^u to a natural exponential function by using the property that b= e^lnb, and then differentiate.