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Questions and Answers
What is the derivative of the function $y = rac{1}{x^2}$ using the natural logarithm differentiation rule?
What is the derivative of the function $y = rac{1}{x^2}$ using the natural logarithm differentiation rule?
Which of the following is a correct application of the chain rule when deriving $y = ext{ln}(u)$ with $u = ext{sin}(x)$?
Which of the following is a correct application of the chain rule when deriving $y = ext{ln}(u)$ with $u = ext{sin}(x)$?
Given the function $y = ext{ln}(x^2 + 1)$, what is the derivative $dy/dx$?
Given the function $y = ext{ln}(x^2 + 1)$, what is the derivative $dy/dx$?
What is the expression for $dy/dx$ given that $y = ext{ln}(e^x)$?
What is the expression for $dy/dx$ given that $y = ext{ln}(e^x)$?
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If $y = ext{ln}( ext{cos}(2x))$, what will be $dy/dx$?
If $y = ext{ln}( ext{cos}(2x))$, what will be $dy/dx$?
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For the function $y = ext{ln}(5x + 3)$, which statement regarding its derivative is correct?
For the function $y = ext{ln}(5x + 3)$, which statement regarding its derivative is correct?
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What is the outcome of differentiating $y = e^{ ext{tan}(x)}$ using the chain rule?
What is the outcome of differentiating $y = e^{ ext{tan}(x)}$ using the chain rule?
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If $y = ext{ln}(x)$, the derivative $dy/dx$ is:
If $y = ext{ln}(x)$, the derivative $dy/dx$ is:
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Study Notes
Derivative of Natural Logarithmic Functions
- A natural logarithm function of x is the logarithm of x to base e (ln|x| = logex).
- x can be rewritten as x = eln x.
- If y = ln x, then d/dx [ln x] = 1/x
- Proof:
- ey = x
- Differentiating both sides: ey (dy/dx) = 1
- dy/dx = 1/ey = 1/x
- Proof:
- Generally, if y = ln[f(x)], then d/dx {ln[f(x)]} = f'(x)/f(x)
Examples of Finding Derivatives
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Example (i): y = ln(x2)
- Let u = x2, so y = ln(u)
- du/dx = 2x, dy/du = 1/u = 1/x2
- dy/dx = (dy/du) * (du/dx) = (1/x2) * (2x) = 2/x
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Example (ii): y = ln(cos2x)
- Let u = cos2x, so y = ln(u)
- du/dx = -2sin(2x), dy/du = 1/u = 1/cos2x = sec2x
- dy/dx = (dy/du) * (du/dx) = (sec2x) * (-2sin(2x)) = -2tan(2x)sec(2x)
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Example (iii): y = ln(√(x2 + 1) / 3√(x3 + 1))
- This example involves more complex substitutions and chain rule application for finding the derivative.
Implicit Differentiation
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Example: y2 - 2y√(1 + x2) + x2 = 0
- Implicitly differentiate both sides with respect to x to find dy/dx.
- The solution involves carefully applying the chain rule and isolating dy/dx to solve for it.
Other Differentiation Exercises (Implicit Differentiation)
- Exercises are given for finding dy/dx for various equations involving x and y (e.g., xy3 - 2x2y2 + x4 = 1, x2sin y - y cos x = 10x3, etc.) These problems require applying implicit differentiation techniques. The solutions involve multiple steps based upon the chain rule and implicit differentiation rules.
Exponential Function Differentiation
- Exercises involve finding derivatives of functions involving exponential functions, trigonometric functions, and logarithms.
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Description
Test your understanding of the derivatives of natural logarithmic functions with this quiz. Explore the rules and examples including the chain rule as applied to ln functions, and practice finding derivatives for various expressions. Perfect for calculus students looking to reinforce their knowledge.
