Calculus Volume 1 · free preview
Derivatives of Inverse Functions
The inverse function theorem turns known derivatives into new ones, delivering the derivatives of the natural logarithm and the inverse trigonometric functions from the derivatives of their inverses.
1. The theorem in action
The theorem in action
2. The derivative of the natural logarithm
The derivative of the natural logarithm
3. Inverse trigonometric derivatives
Inverse trigonometric derivatives
4. Logarithmic differentiation
Logarithmic differentiation
5. Inverse function theorem statement
The inverse function theorem expresses the derivative of f inverse in terms of f prime. Which form is correct?
6. Derivative of the natural log
What is the derivative of f(x) = ln x for positive x?
7. Derivative of arcsin
What is the derivative of f(x) = arcsin x on its domain (-1, 1)?
8. Derivative of arctan
What is the derivative of f(x) = arctan x, and on what domain does the formula hold?
9. Logarithmic differentiation purpose
What is the main advantage of logarithmic differentiation for a complicated product like x^2 sqrt(x + 1) over (x + 2)^3?
10. Endpoint behavior of arcsin
The derivative of arcsin x is 1/sqrt(1 - x^2), which blows up as x approaches 1 or -1. What does this mean geometrically?
11. Derivative of log base ten
The derivative of b^x is b^x ln b. Using change of base, what is the derivative of log base 10 of x? Answer like 1/(x ln 10).
12. Derivative of ln at a point
What is the derivative of f(x) = ln x at x = e? Answer with a fraction like 1/e.
13. Derivative of arcsin at zero
What is the value of the derivative of arcsin x at x = 0? Answer with a number.
14. Sign of arccos derivative
The derivative of arccos x is the negative of the derivative of arcsin x. As a number with a sign, what is the derivative of arccos x at x = 0? Answer with a number.
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