Podcast
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?
- $ rac{1}{x^2}$
- $ rac{-2}{x^4}$
- $- rac{2}{x^3}$ (correct)
- $- rac{1}{2x^2}$
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 $u = ext{sin}(x)$?
- $dy/dx = rac{ ext{cos}(x)}{ ext{sin}(x)}$
- $dy/dx = rac{- ext{cos}(x)}{u^2}$
- $dy/dx = rac{1}{ ext{sin}(x)}$
- $dy/dx = rac{1}{u} rac{du}{dx}$ (correct)
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 $dy/dx$?
- $ rac{x}{x^2 + 2}$
- $ rac{1}{x^2 + 1}$
- $ rac{2x}{x + 1}$
- $ rac{2x}{x^2 + 1}$ (correct)
What is the expression for $dy/dx$ given that $y = ext{ln}(e^x)$?
What is the expression for $dy/dx$ given that $y = ext{ln}(e^x)$?
If $y = ext{ln}( ext{cos}(2x))$, what will be $dy/dx$?
If $y = ext{ln}( ext{cos}(2x))$, what will be $dy/dx$?
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?
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?
If $y = ext{ln}(x)$, the derivative $dy/dx$ is:
If $y = ext{ln}(x)$, the derivative $dy/dx$ is:
Flashcards
Derivative of ln(x)
Derivative of ln(x)
The derivative of the natural logarithm of x is 1/x.
Derivative of ln(f(x))
Derivative of ln(f(x))
The derivative of the natural logarithm of a function f(x) is f'(x) / f(x).
Implicit Differentiation
Implicit Differentiation
Finding the derivative of a function where the variables are not explicitly solved for one another, instead both sides of the equation are differentiated with respect to x, solving for dy/dx
Chain Rule
Chain Rule
Signup and view all the flashcards
Natural Logarithm
Natural Logarithm
Signup and view all the flashcards
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
-
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
-
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)
-
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
- 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.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
