Derivative of a function ：練習問題

Differentiate $$y = 5e^x + \frac{2}{x}$$ with respect to $$x$$.

Given the function $$f(x) = x \ln x$$, find $$f'(x)$$.

Find the slope of the normal to the curve \(y = \frac{x + e^{2x}}{x^2 + 1}\) at the point where it intersects the \(y\)-axis.

Find the derivative of the function $$f(x) = 7x^3 - 4x^2 + 6$$ with respect to $$x$$.

Find the value of \(\lim_{x \to 0} \frac{1}{x} \ln \left( \frac{e^{5x} + e^{7x}}{2} \right)\).

Find the gradient of the tangent to the curve \(y = 4x^2 - 3x + 1\) at the point where \(x = 2\).

Find the value of the constant \(a\) such that the limit \(\lim_{x \to 0} \frac{\sqrt{x+a} - 3}{x}\) exists, and evaluate the limit.

Given that $$y = (x^2 + \ln x)^3 e^{2x}$$, find $$\frac{dy}{dx}$$.

Consider the function
$$f(x) = \frac{\ln(x+1)}{e^{x^2}}$$ for \(x > -1\).

(a) Find the first derivative \(f'(x)\) and express your answer in the form \(e^{-x^2} g(x)\), where \(g(x)\) is a function involving \(x\) and \(\ln(x+1)\).

(b) Find the equation of the tangent line to the curve \(y = f(x)\) at the point where \(x = 0\).

(c) It is given that the second derivative of \(f(x)\) is
$$f''(x) = e^{-x^2} \left[ (4x^3+6x) \ln(x+1) - \frac{4x^2+2x-1}{(x+1)^2} \right]$$ Determine, showing your calculation, whether the curve \(y = f(x)\) is concave up or concave down at \(x = 1\).