Differentiation of a function：練習問題

Find $$\frac{dy}{dx}$$ if $$y = \ln(5x+1)$$

Consider the function \( f(x) = \frac{\ln(2x+1)}{e^{x^2}} \) for \( x > -\frac{1}{2} \). Find the value of \( f'(0) \).

Find the slope of the tangent to the curve \( y = \ln(x^2 + e^{x-1}) \) at the point where \( x = 1 \).

Find \( \frac{dy}{dx} \) if \( y = \frac{4}{x^3} + 2e^{-x} \).

Find $$\frac{dy}{dx}$$ if $$y = x e^{3x}$$

Find the derivative of \(y = x^2 e^{3x}\) with respect to \(x\).

Consider the function \(f(x) = (x^2 + \ln x)^3 e^{1-x}\) for \(x > 0\). Find the value of \(f'(1)\).

Using the quotient rule and the chain rule, find the derivative \(\frac{d}{dx}\left(\frac{\ln(2x)}{x}\right)\) and express your answer in its simplest form.

Consider the function \(f(x) = \frac{x^2}{x-1}\) for \(x \ne 1\).

(a) Find \(f'(x)\).

(b) Determine the coordinates of the stationary points of \(f(x)\).

The population \(P\) of a certain city (in thousands) at time \(t\) years is modeled by the function \(P(t) = \frac{200}{1 + 4e^{-0.2t}}\\, where \)t \ge 0\).

(a) Find the initial population of the city.

(b) Find the rate of change of the population with respect to time, \(\frac{dP}{dt}\).

(c) Find the second derivative \(\frac{d^2P}{dt^2}\).

(d) Determine the time \(t\) at which the population growth rate is maximized.