Exponential Functions and Derivatives in Calculus II Lecture #4
Explore the general exponential functions, properties, derivatives, and integrals in Calculus II lecture #4. Learn about the power rule for derivatives, inverse equations, and logarithms with various bases. Examples provided for a clear understanding of the concepts covered.
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Calculus II Lecture #4 General Exponential Function ?? Civil Engineering Department College of Engineering Mustansiriayah University May 2020 1 Calculus II, Lecture #4 23/04/2025
The General Exponential Function The General Exponential Function ?? Definition: For any numbers a>0 and x, the exponential function with base a is: ??= ?? ln ? Properties of the general exponential function: ax1.ax2= ??1ln ?.??2ln ? = ??1ln ?+?2ln ? = ??1+?2 ln ?= ax1+?2 2 Calculus II, Lecture #4 23/04/2025
General Power Rule for Derivatives General Power Rule for Derivatives If a > 0 and x is a differentiable function, then axis a differentiable function of x and ? ????= ? ???? ln ?= ?? ln ?? ?ln? = ??ln? ?? ?2 ??2??= ? ?? ??ln? = ??ln?2 ? ????= ??ln? ?? ?? 3 Calculus II, Lecture #4 23/04/2025
Example Example 1 1: Find the first derivative for the general exponential functions, ? ??3?= 3?ln3 1. ? ??3 ?= 3 ?ln3 ? ?? ? = 3 ?ln3 2. ? ??3sin ?= 3sin ?ln3 ? ??sin? = 3sin ?ln3 cos? 3. Example 2: Find the first derivative for the function ? = ??,? > 0, Solution: We take xxas a power of e: ? = ??= ?? ln ? ?? ??= ? ???? ln ? = ?? ln ?? ? ln? ?? = ???.1 ?+ ln? = ??1 + ln? 4 Calculus II, Lecture #4 23/04/2025
The integral of The integral of a au u The integral equivalent of this last result gives the general anti-derivative ???? 1 ? ?? ln? ? 1 1 ???? = ???? = ln???+ ? ??= ln? ?? ?? ln?+ ? ???? ??= Example 3: Integrate the following general exponential functions: 2? ln 2+ ? 1. 2??? = 2? ln 2+ ? =2sin ? 2. 2sin ?cos??? = 2??? = ln 2+ ? 5 Calculus II, Lecture #4 23/04/2025
Logarithms with base Logarithms with base a a For any positive number a 1, log?? is the inverse function of ax The graph of the function 2xand its inverse function log2? Inverse equations for axand ????? ? log??= ?, ? > 0 log???= ?, all ? log?? = 1 6 Calculus II, Lecture #4 23/04/2025
Example Example 4 4: Applying the inverse equations: 1. log225= 5 2. log1010 7= 7 3. 2log2(3)= 3 4. 10log10(4)= 4 Evaluation of ????? lnalnx =lnx 1 logax = lna Proof: ? log??= ? ln? log??= ln? log?? .ln? = ln? log?? =ln? ln? 7 Calculus II, Lecture #4 23/04/2025
Example Example 5 5: Determine the values of the following expressions: ln 2 ln 10 0.69315 1. log102 = 2.30259 0.30103 2. logex =ln x ln e=lnx 3. log512 =ln 12 ln 5 2.485 1.609 1.544 4. log723 =ln 23 ln 7 3.1355 1.946 1.611 ln 11 ln 121= ln 11 ln 112= ln 11 2 ln 11=1 5. log12111 = 2 ln 11=ln 112 6. log11121 =ln 121 ln 11=2 ln 11 ln 11= 2 7. log42exsin x=ln 2exsin x =exsin x.ln 2 2 ln 2 =exsin x ln 4 2 8 Calculus II, Lecture #4 23/04/2025
Rules for base ( Rules for base (a) a) logarithms logarithms: : For any x>0 and y>0: Product rule: logaxy =logax +logay Division rule: x y=logax logay loga Reciprocal rule: 1 y= logay loga Power rule: logaxy= ylogax 9 Calculus II, Lecture #4 23/04/2025
Example Example 6 6: Find the values of the following expressions: 1 2= x 1 2log4x= 4log4x 1 2 a) 2log4x= 4 = 5log53x22 b) 25log53x2 = 52 log53x2 = 3x2 2= 9x4 Derivatives and Integrals involving ????? ? ??logau =? ln? ln?= 1 ? ?? ln?.1 1 ?? ?? ln? = ?? ln? ? Example Example 7 7: : ? ??log103x + 1 = b) ? 1 ln2 ? ?? because ? = ln?, 1 1 ? ??3x + 1 = 3 a) ln 10. ln ? ? 3x+1 ?? ln 10 3x+1 log2? 1 ?? = ln 2 ?? =1 = ??? ln2.?2 ln?2 2 ln?2 2ln2+ ? 1 1 = 2+ ? = ln2. + ? = 10 Calculus II, Lecture #4 23/04/2025
