# Differentiating Logarithmic Functions with Base e

Differentiating logarithmic functions with base e involves applying the rules of calculus to functions in the form of ln(x), where ln denotes the natural logarithm with base e. These functions are fundamental in various scientific and mathematical contexts due to their unique properties. Differentiation of such functions requires understanding the derivative rules specific to logarithmic expressions, particularly the chain rule and the derivative of ln(x). Mastery of this concept is crucial in fields like physics, engineering, and finance, where logarithmic functions frequently model phenomena such as growth rates, decay, and complex systems.

Questions

- What is the derivative of #e^x(sinx)#?
- How do I find the derivative of #3e^(-12t) #?
- What is the derivative of #x^(lnx)#?
- What is the derivative of #f(x)=sin(1/lnx)#?
- How do you differentiate #y=12lnx#?
- How do you find the derivative of #y=ln(sqrt(x))#?
- How do I find the derivative of #y=ln((x^2)+1)#?
- How do you find the derivative of exponential function #y = e^9 ln x#?
- What is the derivative of #ln(2x^2+1)#?
- How do you differentiate #y=(sinx)^lnx#?
- What is the derivative of #ln(lnx^2)#?
- What is the derivative of #e^(x-1)#?
- How do you differentiate #y=ln(1/x)-1/lnx#?
- Is there a difference between #lim_(h->0)(f(x+h)-f(x))/h# and #lim_(deltax->0)(f(x+deltax)-f(x))/(deltax)#?
- What is the derivative of # ln(x^2+y^2)#?
- How do you find the derivatives of #z=ln((y^4+1)^2/(y-1)^3)#?
- How do I find the derivative of #y = (sin(e^x))^ln(x^2)#?
- How do I find the derivative of #ln sqrt(x^2-4)#?
- How do you find the derivative of #y= ln ((x^2 (x+1))/ (x+2)^3)#?
- What is the derivative of #ln(x)^x#?