Home

Derivatives of Exp & Log Calculator

Exponential and logarithmic derivative calculator for e^x, a^x, ln x, log_a x, x^x. Enter base and x to get the derivative formula and value—for derivative practice.

展开更多 ▾
(e^x)′=e^x (a^x)′=a^x·ln a (ln x)′=1/x
Exponential or log
Use 1 otherwise
Positive

📖 Tutorial｜Derivatives of Exp & Log

1. Definition
Exp & log derivatives: (e^x)′=e^x, (a^x)′=a^x ln a, (ln x)′=1/x, (log_a x)′=1/(x ln a), (x^x)′=x^x(ln x+1).
2. Symbols
FunctionDerivative
e^xe^x
a^xa^x·ln a
ln x1/x
log_a x1/(x ln a)
x^xx^x(ln x+1)
3. How it works
  • The tool applies the analytic formula;
  • Then substitutes x;
  • x^x comes from logarithmic differentiation.
4. Steps
  1. Choose the function;
  2. Enter the base a if needed;
  3. Enter x and click Calculate.
5. Example
Example: 2^x at x=3. (2^x)′=2^x ln2=8ln2≈5.545.
6. Pitfalls
Do not forget ln a in (a^x)′;
(ln x)′=1/x uses the natural log;
x^x is neither a pure power nor exponential; use log differentiation.

❓ FAQ｜Derivatives of Exp & Log

How is the x^x derivative derived?
Take ln y=x ln x, differentiate y′/y=ln x+1, so y′=x^x(ln x+1).
Why the extra ln a in (a^x)′?
Write a^x=e^(x ln a); the chain rule gives e^(x ln a)·ln a=a^x ln a.
ln x vs log_a x derivatives?
(ln x)′=1/x; (log_a x)′=1/(x ln a), an extra factor 1/ln a.
Why is e^x special?
Its derivative equals itself; the only exponential with this property, by definition of e.
Can x be negative?
ln x, log_a x and x^x require x>0; please enter a positive x.