Calculus Volume 1 · free preview
Derivatives of Exponential and Logarithmic Functions
The exponential reproduces itself under differentiation, the natural logarithm differentiates to the reciprocal, general bases carry a ln b factor, and logarithmic differentiation tames complicated products.
1. The exponential family
The exponential family
2. The logarithm family
The logarithm family
3. Logarithmic differentiation
Logarithmic differentiation
4. Growth and decay models
Growth and decay models
5. Derivative of e^x
What is the derivative of the natural exponential function f(x) = e^x?
6. Derivative of a general base
What is the derivative of the exponential function f(x) = 5^x with base 5?
7. Domain of ln x derivative
The derivative of f(x) = ln x is 1/x. For which x is this formula valid?
8. Variable base and exponent
What is the derivative of y = x^x for positive x?
9. Why absolute values in log differentiation
When logarithmic differentiation writes ln|y| for a product that may be negative, what purpose do the absolute values serve?
10. Derivative of a log composition
What is the derivative of the composition f(x) = ln(x^2 + 1)?
11. Derivative of e^(3x)
What is the derivative of f(x) = e^(3x)? Answer like 3e^(3x).
12. Derivative of ln at e squared
What is the derivative of f(x) = ln x at x = e^2? Answer with a fraction like 1/e^2.
13. Log base 2 derivative
What is the derivative of f(x) = log base 2 of x? Answer like 1/(x ln 2).
14. Coefficient in x to the x derivative
The derivative of x^x is x^x (ln x + 1). At x = 1, what number is this derivative? Answer with a number.
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