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))
- 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) )
- 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} )
- 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
- 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
- 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 )
- 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 )
- 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
- 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 )
- 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] )
- 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
Here are slides 21 to 30:
- 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).
- 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))
- 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).
- 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))
- 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))
- 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})
- 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))
- 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)
- 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])
- 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
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)