Calculus Volume 1 · 免费预览
指数函数与对数函数的导数
指数函数求导后仍为自身，自然对数求导为其倒数，一般底数的指数函数会带有一个 ln b 因子，而对数求导法则能化解复杂的乘积。
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.