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Exploring Exponential Functions: Derivatives, Equations, and Simplifications

This comprehensive guide delves into the intricacies of exponential functions, focusing on finding derivatives, solving equations, and simplifying expressions. Key examples include calculating derivatives of logarithmic forms and employing properties of logarithms to solve complex equations. The review covers essential topics like the differentiation of natural logarithms, manipulation of exponential equations, and practical applications. Master the techniques to navigate challenges in calculus involving exponential functions effectively.

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Exploring Exponential Functions: Derivatives, Equations, and Simplifications

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  1. REVIEW 7-2 Exponential Functions

  2.          3       -----                            3x - 4   Find the derivative:  1. f(x)  =  ln(3x - 4)   2. f(x)  =  ln[(1 + x)(1 + x2)2(1 + x3)3 ] ln(1 + x) + ln(1 + x2)2 + ln(1 + x3)3 ln(1 + x) + 2 ln(1 + x2) + 3 ln(1 + x3)                          1                 4x                   9x2          f '(x)  =   ------     +     --------      +     --------                           1 + x            1 + x2             1 + x3

  3. 3. y = ln(cosx + 8x) -sinx + 8cosx + 8x 4. y = ln(ln12x) 1__x__ln12x 1__xln12x = 5. y = 9xln2x 9x(1/x) + 9ln2x 9 + 9ln2x

  4. 6. y = ex2 7. y = sin(e3x).

  5. SOLVE: 8. ln (x + 4) + ln (x - 2) = ln 7 ln (x + 4)(x - 2) = ln 7 eln (x + 4)(x - 2) = eln 7 (x + 4)(x - 2) = 7 x2 + 2x - 8 = 7 x2 + 2x - 15 = 0 (x - 3)(x + 5) = 0 x = 3 or x = -5

  6. 9. Solve the equation. e3x + 2 = 40 ln e 3x + 2 = ln 40 (3x + 2) ln e = ln 40 Remember that ln e = 1. 3x + 2 = ln 40 3x = ln 40 - 2

  7. 10. Solve for y: ln y2 +3y - ln (y + 3) = 6 y2 + 3yy + 3 ln = 6 ln(y) = 6 y = e6

  8. SIMPLIFY: 11. ln(e3x) 12. e2ln5x 13. eln7x+9 14. ln( ) 3x (5x)2 = 25x2 _1_e2x eln7x + e97xe9 -2x

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