由第一原理求導：练习题

Using differentiation from first principles, show that the derivative of \( f(x) = x^2 \) is \( 2x \). Which of the following represents the correct expression for the gradient of the chord joining the points where the \( x \)-coordinates are \( x \) and \( x + h \)?

A curve has the equation \( y = 3x^2 + 5x \).
Which of the following represents the correct expression for the gradient of the chord joining the points where the \( x \)-coordinates are \( x \) and \( x + h \) before taking the limit \( h \to 0 \)?<\/p>

Using differentiation from first principles, find the derivative of the function \( f(x) = 2x^2 + 3 \).
Which of the following represents the correct expression for the gradient of the chord joining the points on the curve with \( x \)-coordinates \( x \) and \( x + h \)?

A curve has the equation \( y = 3x^4 - 5x + 2 \).
Use differentiation from first principles to find the derivative \( \frac{dy}{dx} \) of the function \( f(x) = x^3 \).

The function \( f(x) \) is defined by \( f(x) = x^2 \).
Use differentiation from first principles to show that \( f'(x) = 2x \).

The function \( f \) is defined by \( f(x) = x^3 - 4x \).
(a) Using differentiation from first principles, show that the derivative \( f'(x) = 3x^2 - 4 \). You must use the definition \( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \).
(b) Hence, find the coordinates of the two stationary points on the curve \( y = f(x) \).
(c) Determine the set of values for which the function \( f(x) \) is decreasing, giving your answer in set notation.