Differentiation：練習問題

Find the derivative of the function \(f(x) = 3\sqrt{x} + 4\cos x\).

Find \(\frac{dy}{dx}\) for the implicit relation \(x^2 + \sin(xy) = 4\).

Given the implicit equation \( \sin y + x \cos y = 1 \), find the value of \( \frac{d^2y}{dx^2} \) at the point \( (1, 0) \).

Differentiate \(f(x) = \frac{1}{x^3} + \sqrt[3]{x^2} - 4\sec x\) with respect to \(x\).

Find \( \frac{dy}{dx} \) if \( y = \ln(\sec x + \tan x) \).

Find the derivative of \( f(x) = \frac{2x - 3}{x + 5} \) with respect to \( x \) for \( x \neq -5 \).

Find the equation of the tangent line to the curve \( y = x^2 - \ln x \) at the point where \( x = 1 \).

Consider the function \(f(x) = (x^2 - 1)e^{ax}\), where \(a\) is a non-zero constant. If the point \(x = 1\) is a point of inflexion of the curve \(y = f(x)\), find the value of \(a\).

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

Consider the curve C defined by the implicit equation: $$\mathrm{e}^y + x = y^2 + xy$$

(a) Show that the first derivative is given by: $$\frac{dy}{dx} = \frac{y - 1}{\mathrm{e}^y - 2y - x}$$

(b) Verify that the point \(P(-1, 0)\) lies on the curve C. Hence, find the equation of the tangent line to C at point P.

(c) Find the value of the second derivative $$\frac{d^2y}{dx^2}$$ at the point \(P(-1, 0)\). Determine whether the curve C is concave up or concave down at P.