Exponential & Logarithmic Derivatives
The exponential function e^x is special: it's its own derivative! This property makes e^x incredibly useful in mathematics and engineering.
Exponential & Log Derivatives
Exponential with Chain Rule
Medium
Find d/dx[e^(x^2)]
1
Apply chain rule
= e^(x^2) * d/dx[x^2] = e^(x^2) * 2x = 2xe^(x^2)
Answer:
2xe^(x^2)
Logarithmic Derivative
Medium
Find d/dx[ln(x^2 + 1)]
1
Apply chain rule
= 1/(x^2+1) * d/dx[x^2+1] = 2x/(x^2+1)
Answer:
2x/(x^2+1)
Practice
1. d/dx[e^(3x)] = ?
2. d/dx[ln(2x)] = ?
Chain rule: (1/(2x)) * 2 = 1/x. Or use ln(2x) = ln(2) + ln(x), then d/dx = 0 + 1/x
Key Takeaways
d/dx[e^x] = e^x (its own derivative!)
d/dx[ln(x)] = 1/x
Always apply chain rule for composite functions
d/dx[a^x] = a^x * ln(a) for other bases