Derivatives - Exponential Functions & its Derivatives

• Today’s topic: Derivatives of Exponential Functions
• What are exponential functions?
• Understanding the general form: (f(x) = a^x), where (a) is a constant
• The derivative of an exponential function
• Example: Find the derivative of (f(x) = 2^x)

Derivatives of Exponential Functions (Cont’d)

• Using the power rule for derivatives
• Derivative of (a^x) with respect to (x)
• Derivative of (e^x) with respect to (x)
• Properties of exponential functions
• Example: Find the derivative of (f(x) = 3e^{2x})

Derivatives of Exponential Functions (Cont’d)

• Derivative of (e^{kx}) with respect to (x)
• Application: Growth and decay problems
• Example: Solve the differential equation (dy/dx = ky)
• Application: Compounding interest problems
• Example: Find the derivative of (P(t) = P_0e^{rt})

Derivatives of Exponential Functions (Cont’d)

• Derivative of (b^x) with respect to (x)
• Example: Find the derivative of (f(x) = 4^x)
• Differentiation rules for exponential functions
• Example: Find the derivative of (g(x) = 10^{2x})
• Summary and Key points

Derivatives - Logarithmic Functions & its Derivatives

• Today’s topic: Derivatives of Logarithmic Functions
• What are logarithmic functions?
• Understanding the general form: (f(x) = \log_a(x)), where (a) is a constant
• The derivative of a logarithmic function
• Example: Find the derivative of (f(x) = \log_2(x))

Derivatives of Logarithmic Functions (Cont’d)

• Using the power rule for derivatives
• Derivative of (\log_a(x)) with respect to (x)
• Derivative of (\ln(x)) with respect to (x)
• Properties of logarithmic functions
• Example: Find the derivative of (f(x) = \ln(5x))

Derivatives of Logarithmic Functions (Cont’d)

• Derivative of (\ln(kx)) with respect to (x)
• Application: Solving equations involving logarithmic functions
• Example: Solve the equation (\ln(2x+1)=3)
• Application: Solving exponential growth and decay problems
• Example: Solve the equation (5e^{2x}=8)

Derivatives of Logarithmic Functions (Cont’d)

• Derivative of (\log_b(x)) with respect to (x)
• Example: Find the derivative of (f(x) = \log_3(x))
• Differentiation rules for logarithmic functions
• Example: Find the derivative of (g(x) = \log_{10}(7x))
• Summary and Key points

Derivatives - Trigonometric Functions & its Derivatives

• Today’s topic: Derivatives of Trigonometric Functions
• Introducing trigonometric functions: (f(x) = \sin(x)), (f(x) = \cos(x)), etc.
• Trigonometric identities and properties
• The derivative of a trigonometric function
• Example: Find the derivative of (f(x) = \sin(2x))

11. Derivatives of Trigonometric Functions (Cont’d)

• Using the power rule for derivatives
• Derivative of ( \sin(x) ) with respect to ( x )
• Derivative of ( \cos(x) ) with respect to ( x )
• Derivative of ( \tan(x) ) with respect to ( x )
• Example: Find the derivative of ( f(x) = 2 \cos(3x) )

12. Derivatives of Trigonometric Functions (Cont’d)

• Derivative of ( \cot(x) ) with respect to ( x )
• Derivative of ( \sec(x) ) with respect to ( x )
• Derivative of ( \csc(x) ) with respect to ( x )
• Application: Solving trigonometric equations
• Example: Solve the equation ( \sin(x) = \frac{1}{2} )

13. Derivatives of Trigonometric Functions (Cont’d)

• Application: Finding tangent lines to trigonometric curves
• Example: Find the equation of the tangent line to ( f(x) = 3 \cos(x) ) at ( x = \frac{\pi}{4} )
• Derivative of inverse trigonometric functions
• Example: Find the derivative of ( f(x) = \sin^{-1}(2x) )
• Derivative of hyperbolic trigonometric functions

14. Derivatives of Trigonometric Functions (Cont’d)

• Applications of hyperbolic trigonometric functions
• Example: Find the derivative of ( f(x) = \sinh(3x) )
• Derivative of inverse hyperbolic trigonometric functions
• Example: Find the derivative of ( f(x) = \tanh^{-1}(2x) )
• Summary and Key points

15. Antiderivatives and Indefinite Integrals

• Introduction to antiderivatives
• Relationship between derivatives and antiderivatives
• Indefinite integrals and the notation ( \int f(x) , dx )
• Basic antiderivative rules
• Example: Find the antiderivative of ( f(x) = 3x^2 )

16. Antiderivatives and Indefinite Integrals (Cont’d)

• Antiderivative of a constant multiple of a function
• Antiderivative of the sum or difference of functions
• Example: Find the antiderivative of ( f(x) = 4x^3 + 2x - 5 )
• Application: Finding areas under curves
• Example: Find the area under the curve ( f(x) = 2x ) between ( x = 1 ) and ( x = 3 )

17. Antiderivatives and Indefinite Integrals (Cont’d)

• Antiderivative of basic trigonometric functions
• Example: Find the antiderivative of ( f(x) = \cos(x) )
• Antiderivative of exponential functions and logarithmic functions
• Example: Find the antiderivative of ( f(x) = e^x )
• Summary and Key points

18. Definite Integrals

• Introduction to definite integrals
• The definite integral symbol and notation ( \int_a^b f(x) , dx )
• Geometric interpretation of definite integrals as areas
• Fundamental Theorem of Calculus
• Example: Evaluate ( \int_0^3 2x , dx )

19. Definite Integrals (Cont’d)

• Properties of definite integrals
• Definite integrals of even and odd functions
• Example: Evaluate ( \int_{-2}^2 x^3 , dx )
• Application: Finding the average value of a function
• Example: Find the average value of ( f(x) = x^2 ) on the interval ( [0, 2] )

20. Definite Integrals (Cont’d)

• Application: Finding displacement and velocity
• Example: Find the displacement and velocity of an object, given its acceleration function ( a(t) = 2t )
• Summary and Key points

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21. Derivatives - Exponential Functions & its Derivatives (Cont’d)

• Exponential growth and decay problems (cont’d)
• Example: A bacteria population grows according to the equation (P(t) = 100e^{0.05t}). Find the rate of growth at (t = 3).
• Compounding interest problems (cont’d)
• Example: An investment grows according to the equation (A(t) = P_0(1 + r)^t). Find the rate of change of the investment at (t = 5).

22. Derivatives - Exponential Functions & its Derivatives (Cont’d)

• Properties of logarithmic functions (cont’d)
• Example: Simplify the expression (\log_2(16) - \log_2(4))
• Solving equations involving logarithmic functions
• Example: Solve the equation (\log(x+1) = \log(3x))

23. Derivatives - Exponential Functions & its Derivatives (Cont’d)

• Exponential growth and decay problems using logarithmic functions
• Example: A radioactive substance decays according to the equation (Q(t) = Q_0e^{-kt}). Find the half-life of the substance.
• Finding tangent lines to exponential curves
• Example: Find the equation of the tangent line to (f(x) = 5e^x) at (x = 2).

24. Derivatives - Trigonometric Functions & its Derivatives (Cont’d)

• Derivative of inverse trigonometric functions (cont’d)
• Example: Find the derivative of (f(x) = \tan^{-1}(2x))
• Derivative of hyperbolic trigonometric functions (cont’d)
• Example: Find the derivative of (f(x) = \sinh(5x))

25. Derivatives - Trigonometric Functions & its Derivatives (Cont’d)

• Applications of hyperbolic trigonometric functions (cont’d)
• Example: Find the derivative of (f(x) = \cosh(3x))
• Derivative of inverse hyperbolic trigonometric functions (cont’d)
• Example: Find the derivative of (f(x) = \tanh^{-1}(4x))

26. Antiderivatives and Indefinite Integrals (Cont’d)

• Antiderivative of higher powers of (x)
• Example: Find the antiderivative of (f(x) = x^4)
• Antiderivative of rational functions
• Example: Find the antiderivative of (f(x) = \frac{2}{x})

27. Antiderivatives and Indefinite Integrals (Cont’d)

• Antiderivative of trigonometric functions
• Example: Find the antiderivative of (f(x) = \sin(2x))
• Antiderivative of exponential functions and logarithmic functions (cont’d)
• Example: Find the antiderivative of (f(x) = \ln(x))

28. Definite Integrals (Cont’d)

• The definite integral as the limit of Riemann sums
• Example: Evaluate (\int_0^4 3x , dx)
• Fundamental Theorem of Calculus (cont’d)
• Example: Find the derivative of the function (F(x) = \int_0^x \sin(t) , dt)

29. Definite Integrals (Cont’d)

• Properties of definite integrals (cont’d)
• Example: Evaluate (\int_{-1}^1 (x^2 - 2x + 3) , dx)
• Application: Finding the area between curves
• Example: Find the area between the curves (y = x^2) and (y = 2x) on the interval ([-1, 2])

30. Definite Integrals (Cont’d)

• Application: Finding work and fluid force
• Example: Find the work done by a force field (F(x) = 2x - 3) in moving an object along the interval ([-1, 3])
• Summary and Key points