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# OCR A-level Maths Differentiation and graphs The gradient of a curve is given by the derivative of the equation of the curve. The second derivative of a function f(x), given by f''(x), and the second derivative of an equation y, given by d²y/dx², can be found by differentiating the function and the equation twice. For graphs, the second derivative represents the rate of change of the gradient. At the stationary points and points of inflection of a graph, the derivative of the function of the equation of the graph is 0. Therefore, these points can be found by finding the derivative of the function, equating it to 0 and solving for x. The maxima of a graph are the high points of the graph at which the slope is zero. At maxima, the second derivative of the function is less than 0. The minima of a graph are the low points of the graph at which the slope is zero. At minima, the second derivative of the function is more than 0. A function f(x) is said to be increasing at point x if f'(x) > 0 or decreasing at point x if f'(x) < 0. # ✅

Given that the equation of a curve is y = x(cosx +x²), find the gradient of the curve. # ✅

Find the gradient of the curve y = 1/(x³ + 18x) at x = 1. # ✅

How is the second derivative of a function f(x) found and how is it denoted? # ✅

Find the rate of change of the gradient of the graph y = 18xsinx. # ✅

Find the second derivative of the function f(x) = (cos²x + 7x)². # ✅

Use differentiation to show that the graph given by the equation y = 5/(x + 17) does not have any stationary or inflection points. # ✅

Is (2, 125) a stationary point of the graph y = (1 + x²)³? # ✅

Check whether (1, 19) is a stationary point of the graph given by the equation y = x³ + 8x + 10. # ✅

Given that (2, 10) is a minimum point of a graph given by the function f(x), evaluate f'(2). # ✅

Find the stationary points of y = x³/3 − x²/2 − 6x and classify them as maxima and minima. # ✅

Find whether the curve y = 4x² − x³ at x = 3 Is increasing or decreasing. # ✅

Is the function f(x) = tanx + 5x³ increasing or decreasing at x = −3? # ✅

What is the interval at which y = −x³ + 12x + 50 is decreasing?  Have you found the questions useful?