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Questions and Answers
What is the derivative of the inverse function for f(x) = -3x + 3?
What is the derivative of the inverse function for f(x) = -3x + 3?
- -1 (correct)
- -1/5
- -1/3
- -3
For the function f(x) = -5x + 1, what is (f^{-1})' (x)?
For the function f(x) = -5x + 1, what is (f^{-1})' (x)?
- -1
- -5
- -1/6
- -1/5 (correct)
What is the formula used to find (f^{-1})' (x)?
What is the formula used to find (f^{-1})' (x)?
- dy/dx
- 0
- dx/dy
- 1/f'(f^{-1}(x)) (correct)
For the function f(x) = -2x + 2, what is the slope of its inverse at a given x?
For the function f(x) = -2x + 2, what is the slope of its inverse at a given x?
What is the derivative (f^{−1})' (x) for f(x) = x^7 + x - 3?
What is the derivative (f^{−1})' (x) for f(x) = x^7 + x - 3?
If f(x) = -4x^3 - 4, what is the derivative of the inverse f^{-1})' (x)?
If f(x) = -4x^3 - 4, what is the derivative of the inverse f^{-1})' (x)?
What is the significance of the negative sign in the inverse function derivative?
What is the significance of the negative sign in the inverse function derivative?
For the function f(x) = -2x - 3, what is (f^{-1})' (x)?
For the function f(x) = -2x - 3, what is (f^{-1})' (x)?
What is the derivative of the function f(x) = -2x + 3?
What is the derivative of the function f(x) = -2x + 3?
What is the derivative of the function $f(x) = -3x + 3$?
What is the derivative of the function $f(x) = -3x + 3$?
Using the theorem, what is $(f^{-1})'(x)$ for the function $f(x) = -2x + 3$?
Using the theorem, what is $(f^{-1})'(x)$ for the function $f(x) = -2x + 3$?
What is the derivative of the function $f(x) = -5x + 1$?
What is the derivative of the function $f(x) = -5x + 1$?
For the function $f(x) = -2x + 2$, what is its derivative?
For the function $f(x) = -2x + 2$, what is its derivative?
Using the formula $dy = rac{1}{dx}$ for $y = f(x)$, find $(f^{-1})'(x)$ for $f(x) = -2x - 3$.
Using the formula $dy = rac{1}{dx}$ for $y = f(x)$, find $(f^{-1})'(x)$ for $f(x) = -2x - 3$.
What is the derivative of the function $f(x) = -4x^3 - 4$?
What is the derivative of the function $f(x) = -4x^3 - 4$?
For the function $f(x) = x^7 + x - 3$, what is $f'(x)$?
For the function $f(x) = x^7 + x - 3$, what is $f'(x)$?
What is the derivative of the function $f(x) = 3x^5 + 2x + 5$?
What is the derivative of the function $f(x) = 3x^5 + 2x + 5$?
Using the inverse derivative theorem, find $(f^{-1})'(x)$ for $f(x) = -4x + 2$.
Using the inverse derivative theorem, find $(f^{-1})'(x)$ for $f(x) = -4x + 2$.
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Study Notes
Derivatives of Inverse Functions
- The derivative of an inverse function can be found directly (by computation) or through a theorem or formula.
Direct Computation
- When finding the derivative of an inverse function directly, you can solve for the inverse function first and then differentiate it.
Theorem (f^-1)'(x) = 1 / f'(f^-1(x))
- This theorem provides a formula that can be used to find the derivative of an inverse function.
- The formula uses the first derivative of the original function at the inverse function, which is f^-1(x).
Inverse Function Formula dy = 1 / (dx / dy)
- This formula establishes a relationship between the derivative of the original function and the derivative of its inverse function.
- You can find the derivative of an inverse function by first expressing the original function in terms of y = f(x), then differentiating both sides with respect to y, and finally expressing the result in terms of x.
Examples
- Examples illustrate the application of different methods to find the derivative of an inverse function.
- The methods include direct computation, using the theorem, and using the inverse function formula.
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