Cambridge OCR A Level · Mathematics A - H240
3 multiple-choice questions marked as you go, and 3 written questions with worked solutions. All on Differentiation from first principles.
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 \( f(x) = x^3 \).
Write your answer out first, then check it against the worked solution.
The function \( f(x) \) is defined by \( f(x) = x^2 \).
Use differentiation from first principles to show that \( f'(x) = 2x \).
Write your answer out first, then check it against the worked solution.
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.
Write your answer out first, then check it against the worked solution.
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