Derivatives - Logarithmic Differentiation
- Logarithmic differentiation is a technique used to differentiate functions that are products, quotients, or powers involving logarithmic functions.
- It is particularly useful when dealing with complicated functions that cannot be easily differentiated using basic differentiation techniques.
- This method involves taking the natural logarithm of both sides of an equation and then differentiating implicitly.
- Let’s see how this technique can be applied to various types of functions.
- Example: Differentiate the function y = (2x^3 + 3x^2 + 4x + 5) / (e^x)
Logarithmic Differentiation Technique
- Start by taking the natural logarithm of both sides of the equation.
- Example: ln(y) = ln((2x^3 + 3x^2 + 4x + 5) / (e^x))
- Apply the logarithm properties to simplify the equation.
- Example: ln(y) = ln(2x^3 + 3x^2 + 4x + 5) - ln(e^x) = ln(2x^3 + 3x^2 + 4x + 5) - x
- Differentiate both sides of the equation implicitly.
- Example: d/dx(ln(y)) = d/dx(ln(2x^3 + 3x^2 + 4x + 5) - x)
- Use the chain rule to differentiate the natural logarithm of y.
- Example: (1/y) * dy/dx = (1/(2x^3 + 3x^2 + 4x + 5)) * (6x^2 + 6x + 4) - 1
- Solve for dy/dx, which is the derivative of the original function.
- Example: dy/dx = y * [(6x^2 + 6x + 4) / (2x^3 + 3x^2 + 4x + 5)] - y
Logarithmic Differentiation Examples
- Example: Differentiate the function y = x^x
- Apply logarithmic differentiation technique to simplify the expression: ln(y) = ln(x^x) ln(y) = x * ln(x)
- Differentiate implicitly: (1/y) * dy/dx = 1 * ln(x) + x * (1/x) dy/dx = y * [ln(x) + 1]
- Example: Differentiate the function y = (sin(x))^cos(x)
- Apply logarithmic differentiation technique: ln(y) = ln((sin(x))^cos(x)) ln(y) = cos(x) * ln(sin(x))
- Differentiate implicitly: (1/y) * dy/dx = -sin(x) * ln(sin(x)) + cos(x) * (1/sin(x)) * cos(x) dy/dx = y * (-sin(x) * ln(sin(x)) + cos^2(x) / sin(x))
- Example: Differentiate the function y = x^(x^x)
- Apply logarithmic differentiation technique: ln(y) = ln(x^(x^x)) ln(y) = x^x * ln(x)
- Differentiate implicitly: (1/y) * dy/dx = (x^x) * (1/x) * ln(x) + (x^x) * (ln(x) + x * (1/x)) dy/dx = y * (x^x / x * ln(x) + x^x * (ln(x) + 1)) These were examples of logarithmic differentiation applied to different types of functions. Practice using this technique to solve more complex functions.
Slide 11: Logarithmic Differentiation Technique (contd.)
- Example: Differentiate the function y = x^2 / sqrt(x+1)
- Apply logarithmic differentiation technique:
- ln(y) = ln(x^2 / sqrt(x+1))
- ln(y) = 2ln(x) - (1/2)ln(x+1)
- Differentiate implicitly:
- (1/y) * dy/dx = 2 * (1/x) - (1/2) * (1/(x+1))
- dy/dx = y * [2/x - 1/(2(x+1))]
- Apply logarithmic differentiation technique:
- Example: Differentiate the function y = (x^3 + 1)^x / e^(2x)
- Apply logarithmic differentiation technique:
- ln(y) = ln((x^3 + 1)^x / e^(2x))
- ln(y) = x * ln(x^3 + 1) - 2x
- Differentiate implicitly:
- (1/y) * dy/dx = ln(x^3 + 1) + x * (3x^2 / (x^3 + 1)) - 2
- dy/dx = y * [ln(x^3 + 1) + (3x^3 / (x^3 + 1)) - 2]
- Apply logarithmic differentiation technique:
- Example: Differentiate the function y = ln(sqrt(x)) / cos(x)
- Apply logarithmic differentiation technique:
- ln(y) = ln(ln(sqrt(x)) / cos(x))
- ln(y) = ln(ln(sqrt(x))) - ln(cos(x))
- Differentiate implicitly:
- (1/y) * dy/dx = (1/ln(sqrt(x))) * (1/(2sqrt(x))) - (-tan(x))
- dy/dx = y * [(1/(ln(sqrt(x)))) * (1/(2sqrt(x))) + tan(x)]
- Apply logarithmic differentiation technique:
Slide 12: Logarithmic Differentiation Examples
- Example: Differentiate the function y = e^x * ln(e^x) / cos^2(x)
- Apply logarithmic differentiation technique:
- ln(y) = ln(e^x * ln(e^x)) - 2ln(cos(x))
- ln(y) = x + ln(ln(e^x)) - 2ln(cos(x))
- Differentiate implicitly:
- (1/y) * dy/dx = 1 + (1/ln(e^x)) * (1/e^x) - 2 * (-tan(x)/cos(x))
- dy/dx = y * [1 + (1/(ln(e^x))) * (1/e^x) + 2 * (tan(x)/cos(x))]
- Apply logarithmic differentiation technique:
- Example: Differentiate the function y = (log(x))^2 * sqrt(log(x))
- Apply logarithmic differentiation technique:
- ln(y) = ln((log(x))^2 * sqrt(log(x)))
- ln(y) = 2ln(log(x)) + (1/2)ln(log(x))
- Differentiate implicitly:
- (1/y) * dy/dx = (2 * (1/(log(x))) * (1/x)) + (1/2) * (1/(x * log(x)))
- dy/dx = y * [2/(x * log(x)) + 1/(2x * log(x))]
- Apply logarithmic differentiation technique:
- Example: Differentiate the function y = (e^x * ln(x))^2 / x^3
- Apply logarithmic differentiation technique:
- ln(y) = ln((e^x * ln(x))^2 / x^3)
- ln(y) = 2ln(e^x * ln(x)) - 3ln(x)
- Differentiate implicitly:
- (1/y) * dy/dx = (2 * (1/(e^x * ln(x))) * (e^x * ln(x) + e^x/x)) - (3 * (1/x))
- dy/dx = y * [2 * (e^x + 1/x) - (3/x)]
- Apply logarithmic differentiation technique:
Add more slides as needed Derivatives - Logarithmic Differentiation
Slide 21:
- Example: Differentiate the function y = e^(2x) * ln(3x)
- Apply logarithmic differentiation technique:
- ln(y) = ln(e^(2x) * ln(3x))
- ln(y) = 2x + ln(ln(3x))
- Differentiate implicitly:
- (1/y) * dy/dx = 2 + (1/(ln(3x))) * (1/(3x)) * 3
- dy/dx = y * [2 + 1/(3x * ln(3x))]
- Apply logarithmic differentiation technique:
Slide 22:
- Example: Differentiate the function y = (log₂(x))^2 / x^3
- Apply logarithmic differentiation technique:
- ln(y) = ln((log₂(x))^2 / x^3)
- ln(y) = 2ln(log₂(x)) - 3ln(x)
- Differentiate implicitly:
- (1/y) * dy/dx = (2 * (1/(log₂(x))) * (1/(x * ln(2)))) - (3 * (1/x))
- dy/dx = y * [2/(x * ln(2) * log₂(x)) - 3/x]
- Apply logarithmic differentiation technique:
Slide 23:
- Example: Differentiate the function y = (x^n) / (logₐ(x))
- Apply logarithmic differentiation technique:
- ln(y) = ln((x^n) / (logₐ(x)))
- ln(y) = n * ln(x) - ln(logₐ(x))
- Differentiate implicitly:
- (1/y) * dy/dx = (n * (1/x)) - (1/(x * ln(a) * logₐ(x)))
- dy/dx = y * [n/x - 1/(x * ln(a) * logₐ(x))]
- Apply logarithmic differentiation technique:
Slide 24:
- Example: Differentiate the function y = (2^(2x) - 3^x) / (log(x)^2)
- Apply logarithmic differentiation technique:
- ln(y) = ln((2^(2x) - 3^x) / (log(x)^2))
- ln(y) = ln((2^(2x) - 3^x)) - 2ln(log(x))
- Differentiate implicitly:
- (1/y) * dy/dx = (2^(2x) * ln(2) - 3^x * ln(3)) - (2 * (1/x))
- dy/dx = y * [(2^(2x) * ln(2) - 3^x * ln(3)) / (x * log(x))]
- Apply logarithmic differentiation technique:
Slide 25:
- Example: Differentiate the function y = (e^(x^2) - ln(x^3)) / x
- Apply logarithmic differentiation technique:
- ln(y) = ln((e^(x^2) - ln(x^3)) / x)
- ln(y) = ln((e^(x^2) - ln(x^3))) - ln(x)
- Differentiate implicitly:
- (1/y) * dy/dx = (2x * e^(x^2) - (3/x^3)) - (1/x)
- dy/dx = y * [(2x * e^(x^2) - (3/x^3)) / x]
- Apply logarithmic differentiation technique:
Slide 26:
- Example: Differentiate the function y = ((ln(x))^2 - e^x * cos(x)) / x^2
- Apply logarithmic differentiation technique:
- ln(y) = ln(((ln(x))^2 - e^x * cos(x)) / x^2)
- ln(y) = 2ln(ln(x)) - ln(e^x * cos(x)) - 2ln(x)
- Differentiate implicitly:
- (1/y) * dy/dx = (2 * (1/(ln(x))) * (1/x)) - (e^x * cos(x) + e^x * sin(x)) - (2 * (1/x))
- dy/dx = y * [2/(x * ln(x)) - (e^x * cos(x) + e^x * sin(x)) / x]
- Apply logarithmic differentiation technique:
Slide 27:
- Example: Differentiate the function y = (2^(x^2) - tan(x)) / (log(x) * cos(x))
- Apply logarithmic differentiation technique:
- ln(y) = ln((2^(x^2) - tan(x)) / (log(x) * cos(x)))
- ln(y) = ln((2^(x^2) - tan(x))) - ln(log(x) * cos(x))
- Differentiate implicitly:
- (1/y) * dy/dx = (2^(x^2) * ln(2) - sec^2(x) - (1/(x * ln(x))) - (tan(x) * cot(x)))
- dy/dx = y * [(2^(x^2) * ln(2) - sec^2(x) - (1/(x * ln(x))) - (tan(x) * cot(x)))]
- Apply logarithmic differentiation technique:
Slide 28:
- Example: Differentiate the function y = (log₂(x) - ln(x)) / sin(x)
- Apply logarithmic differentiation technique:
- ln(y) = ln((log₂(x) - ln(x)) / sin(x))
- ln(y) = ln((log₂(x) - ln(x))) - ln(sin(x))
- Differentiate implicitly:
- (1/y) * dy/dx = ((1/(x * ln(2))) - (1/x)) - (1/sin(x)) * cos(x)
- dy/dx = y * [(1/(x * ln(2))) - (1/x) - (cos(x)/sin(x))]
- Apply logarithmic differentiation technique:
Slide 29:
- Example: Differentiate the function y = (e^x * ln(x)) / (cos(x) * sin(x))
- Apply logarithmic differentiation technique:
- ln(y) = ln((e^x * ln(x)) / (cos(x) * sin(x)))
- ln(y) = ln(e^x * ln(x)) - ln(cos(x) * sin(x))
- Differentiate implicitly:
- (1/y) * dy/dx = (e^x * (1/x) + (1/x)) - ((cos(x) * sin(x)) + (cos(x) * cos(x) - sin(x) * sin(x)) / (cos(x) * sin(x)))
- dy/dx = y * [(e^x * (1/x) + (1/x)) - (cos(x) * cos(x) + sin(x) * sin(x)) / (cos(x) * sin(x)))]
- Apply logarithmic differentiation technique:
Slide 30:
- Example: Differentiate the function y = (log(x) * sin(x)) / (e^x * cos(x))
- Apply logarithmic differentiation technique:
- ln(y) = ln((log(x) * sin(x)) / (e^x * cos(x)))
- ln(y) = ln(log(x) * sin(x)) - ln(e^x * cos(x))
- Differentiate implicitly:
- (1/y) * dy/dx = ((1/x) * sin(x) + log(x) * cos(x)) - ((e^x * sin(x) + e^x * cos(x) / (e^x * cos(x)))
- dy/dx = y * [((1/x) * sin(x) + log(x) * cos(x)) - (e^x * sin(x) + e^x * cos(x) / (e^x * cos(x)))]
- Apply logarithmic differentiation technique:
Derivatives - Logarithmic Differentiation Logarithmic differentiation is a technique used to differentiate functions that are products, quotients, or powers involving logarithmic functions. It is particularly useful when dealing with complicated functions that cannot be easily differentiated using basic differentiation techniques. This method involves taking the natural logarithm of both sides of an equation and then differentiating implicitly. Let’s see how this technique can be applied to various types of functions. Example: Differentiate the function y = (2x^3 + 3x^2 + 4x + 5) / (e^x)