Differentiation — AS & A Level Mathematics (9709) questions by topic

The equation of a curve is such that dy/dx = 4(2x − 5)^3 − 9x^(1/2). The curve passes through the point A(4, −11/2). Find the gradient of the normal to the curve at the point A.

The equation of a curve is y = 4x^2 + 9/(x^2) − 8. A point P is moving along the curve in such a way that its y-coordinate is decreasing at 5 units per second. Find the rate at…

A point P is moving along the curve with equation y = a x^(3/2) − 12x in such a way that the x-coordinate of P is increasing at a constant rate of 5 units per second. Find the…

A curve has equation y = 2x + 12/(x^2). Find the equation of the tangent to the curve at the point (−2, −1). Give your answer in the form y = mx + c.

A curve is such that dy/dx = 3x² + 10x − 8. Find the set of values of x for which y decreases as x increases.

A curve C has equation y = 9/(2x − 5) + 2x − 5. Find the coordinates of the two stationary points.

The equation of a curve is y = x^3 + ax^2 + bx + 5. The curve has a stationary point at (1, 9). Find the values of the constants a and b. [5]

Given that y = 6x cos(x^2 + 1), find an expression for dy/dx. [2]

The diagram shows the curve with equation y = 6e^(2x) − e^(3x). The shaded region is bounded by the axes and the curve. Find the exact x-coordinate of the maximum point. [3]

The parametric equations of a curve are x = (2t + 1)/(3t + 4), y = 2 ln(3t + 4), where t −4/3. Show that dy/dx can be expressed in the form c(3t + 4) and state the value of the…

Find the coordinates of the stationary points of the curve with equation y = 8x/(2x + 3) − 6x + 5.

Find the coordinates of the stationary points of the curve with equation y = 8x/(2x + 3) − 6x + 5.

Find the coordinates of the stationary points of the curve with equation y = 8x/(2x + 3) − 6x + 5.

A curve has equation (x^2 − 3) ln y + 6x = 14. Show that there is no point on the curve at which the y-coordinate is e^(−1).

The parametric equations of a curve are x = e^(tan t), y = 3 tan² t. Find the equation of the tangent to the curve at the point (e, 3). Give your answer in the form y = mx + c,…

The diagram shows the curve y = cos x √(sin 2x) for 0 ⩽ x ⩽ (1/2)π. The curve has a maximum point at M, where x = a. Find the exact value of a.

The diagram shows the graph of y = 5 sin 2x cos^2 x for 0 <= x <= (1/2)pi and its maximum point M. Find the exact x-coordinate of M. [6]

The equation of a curve is xy + y² e^(−x) = 4. Show that dy/dx = (y² − y e^x) / (x e^x + 2y).

The diagram shows the curve y = √x sin 2x for 0 ⩽ x ⩽ (1/2)π. The curve has a maximum point at M, where x = a. Show that tan 2a = −4a.

Find the exact coordinates of the stationary point of the curve with equation y = 3x^3 ln(x^4), for x 0. [5]

The parametric equations of a curve are x = 2/(cos 3t) and y = tan 3t, for 0 <= t <= 2π. Show that dy/dx can be written as A cosec 3t, where A is a constant to be found. [5]

The equation of a curve is y = tan^(−1)(4x). Find the exact values of x when the gradient of the curve is 1/4. [3]

The curve with equation y = 2x − 8x^(1/2) has a minimum point at A and intersects the positive x-axis at B. Find the coordinates of A and B.

A function f is such that f′(x) = 6(2x − 3)² − 6x for x ∈ ℝ. Determine the set of values of x for which f(x) is decreasing.

The equation of a curve is y = (5 − 2x)^(3/2) + 5 for x < 5/2. A point P is moving along the curve in such a way that the y-coordinate of point P is decreasing at 5 units per…

The equation of a curve is y = 2x^2 − 1/(2x) + 3. Find the coordinates of the stationary point.

The diagram shows the curve with equation y = √(2x^3 + 10). Find the equation of the tangent to the curve at the point where x = 3. Give your answer in the form ax + by + c = 0…

A curve has equation y = 2 tan x − 5 sin x for 0 ≤ x < (1/2)π. Find the x-coordinate of the stationary point of the curve. Give your answer correct to 3 significant figures. [3]

A curve has equation x² ln y + y² + 4x = 9. Find the gradient of the curve at the point (2, 1). [5]

A curve has equation y = (1 + e^(2x))/(1 + 3x). The curve has exactly one stationary point P. Find dy/dx and hence show that the x-coordinate of P satisfies the equation x = 1/6 +…

The curve with equation y = 8e^(−x) − e^(2x) crosses the y-axis at the point A. Find the gradient of the curve at A.

A curve is defined by the parametric equations x = 4cos²t, y = √3 sin 2t, for values of t such that 0 < t < ½π. Find the equation of the normal to the curve at the point for which…

The curve with equation y = ln(2x + 1)/(x + 3) has a maximum point M. Find an expression for dy/dx.

The diagram shows the curve with equation y = 8e^(−x) − e^(2x). The curve crosses the y-axis at the point A and the x-axis at the point B. The shaded region is bounded by the…

A curve is defined by the parametric equations x = 4cos²t, y = √3 sin 2t, for values of t such that 0 < t < ½π. Find the equation of the normal to the curve at the point for which…

The diagram shows the curve with equation y = ln(2x + 1) / (x + 3). The curve has a maximum point M. Find an expression for dy/dx.

The equation of a curve is y = e^(sin x) / cos²x for 0 ≤ x ≤ 2π. Find dy/dx and hence find the x-coordinates of the stationary points of the curve.

Given that 2x = tan y, show that dy/dx = 2 / (1 + 4x²).

The equation of a curve is ye^(2x) + y^2 e^x = 6. Find the gradient of the curve at the point where y = 1. [6]

The diagram shows the curve y = xe^(-ax), where a is a positive constant, and its maximum point M. Find the exact coordinates of M. [4]

By writing y = sec^3 theta as 1/(cos^3 theta), show that dy/dtheta = 3 sin theta sec^4 theta. [2]

Find the exact coordinates of the stationary point of the curve y = e^(2x) sin 2x for 0 ⩽ x ⩽ ½π.

The curve y = x² − a/x has a stationary point at (−3, b). Find the values of the constants a and b.

The equation of a curve is such that dy/dx = 4x − 3√x + 1. Find the x-coordinate of the point on the curve at which the gradient is 11/2.

The diagram shows part of the curve with equation y = 12/∛(2x + 1). The point A on the curve has coordinates (7/2, 6). Find the equation of the tangent to the curve at A. Give…

The equation of a curve is y = 4 + 5x + 6x² − 3x³. Find the set of values of x for which y decreases as x increases.

The equation of a curve is y = 2x² − 3. Two points A and B with x-coordinates 2 and (2 + h) respectively lie on the curve. Find and simplify an expression for the gradient of the…

A function f with domain x 0 is such that f′(x) = 8(2x − 3)^(1/3) − 10x^(2/3). It is given that the curve with equation y = f(x) passes through the point (1, 0). Find the equation…

The equation of a curve is y = kx^(1/2) − 4x² + 2, where k is a constant. Find dy/dx and d²y/dx² in terms of k.

The function f is defined by f(x) = tan²((1/2)x) for 0 ≤ x < π. Find the exact value of f'((2/3)π). [3]

A curve has parametric equations x = (e^(2t) − 2)/(e^(2t) + 1), y = e^(3t) + 1. Find an expression for dy/dx in terms of t. [4]

Let f(x) = 4 sin²3x. Find the value of f'(¼π).

A curve has equation 6e^(−x)y² + e^(2x) − 12y + 7 = 0. Find the gradient of the curve at the point (ln 3, 2).

The function f is defined by f(x) = tan²(½x) for 0 ≤ x < π. Find the exact value of f′(⅔π).

A curve has parametric equations x = (e^(2t) − 2)/(e^(2t) + 1), y = e^(3t) + 1. Find an expression for dy/dx in terms of t.

The equation of a curve is ln(x + y) = 3x²y. Find the gradient of the curve at the point (1, 0).

The diagram shows the curve y = sin 2x(1 + sin 2x), for 0 ≤ x ≤ (3/4)π, and its minimum point M. The shaded region bounded by the curve that lies above the x-axis and the x-axis…

The parametric equations of a curve are x = tan²2t, y = cos 2t, for 0 < t < ¼π. Show that dy/dx = −½ cos³2t.

Let f(x) = 2e^(2x) / (e^(2x) − 3e^x + 2). Find f′(x) and hence find the exact coordinates of the stationary point of the curve with equation y = f(x).

The parametric equations of a curve are x = 3 sin 2t, y = tan t + cot t, for 0 < t < (1/2)π. Show that dy/dx = −2 / (3 sin²2t).

The diagram shows the curve y = 2 sin x √(2 + cos x), for 0 ≤ x ≤ 2π, and its minimum point M, where x = a. Find the value of a correct to 2 decimal places.

Water is poured into a tank at a constant rate of 500 cm³ per second. The depth of water is h cm at time t seconds and the volume is V = (4/3)(25 + h)³ − 62500/3 cm³. Find the…

The equation of a curve is such that dy/dx = 6x² − 30x + 6a, where a is a positive constant. The curve has a stationary point at (a, −15). Find the value of a.

The equation of a curve is y = k√(4x + 1) − x + 5, where k is a positive constant. Find dy/dx.

The diagram shows the points A (1½, 5½) and B (7½, 3½) lying on the curve with equation y = 9x − (2x + 1)^(3/2). Find the coordinates of the maximum point of the curve.

A curve has equation y = (2 + 3 ln x) / (1 + 2x). Find the equation of the tangent to the curve at the point (1, 2/3). Give your answer in the form ax + by + c = 0, where a, b and…

The diagram shows the curve with parametric equations x = 4e^(2t), y = 5e^(−t) cos 2t, for −(1/4)π ≤ t ≤ (1/4)π. The curve has a maximum point M. Find an expression for dy/dx in…

The diagram shows the curve with equation y = e^(−x/2)(x² − 5x + 4). The curve crosses the x-axis at the points A and B, and has a maximum at the point C. Find the exact gradient…

A curve has parametric equations x = (2t + 3)/(t + 2), y = t² + at + 1, where a is a constant. It is given that, at the point P on the curve, the gradient is 1. Show that the…

The diagram shows the curve with equation y = e^(−½x)(x² − 5x + 4). The curve crosses the x-axis at the points A and B, and has a maximum at the point C. Find the exact gradient…

A curve has parametric equations x = (2t + 3) / (t + 2), y = t² + at + 1, where a is a constant. It is given that, at the point P on the curve, the gradient is 1. Show that the…

The equation of a curve is x²y − ay² = 4a³, where a is a non-zero constant. Show that dy/dx = 2xy / (2ay − x²).

The equation of a curve is 3x² + 4xy + 3y² = 5. Show that dy/dx = −(3x + 2y)/(2x + 3y).

The diagram shows the curve y = (x + 5)√(3 − 2x) and its maximum point M. Find the exact coordinates of M.

The parametric equations of a curve are x = cos θ / (2 − sin θ), y = θ + 2 cos θ. Show that dy/dx = (2 − sin θ)².

The diagram shows the part of the curve y = x² cos 3x for 0 ⩽ x ⩽ ⅙π, and its maximum point M, where x = a. Show that a satisfies the equation a = ⅓ tan⁻¹(2/(3a)).

The diagram shows a cubical closed container made of a thin elastic material which is filled with water and frozen. During the freezing process the length, x cm, of each edge of…

The equation of a curve is such that dy/dx = (1/2)x + 72/x⁴. The curve passes through the point P(2, 8). Find the equation of the normal to the curve at P.

The equation of a curve is y = f(x), where f(x) = (4x − 3)^(5/3) − (20/3)x. Find the x-coordinates of the stationary points of the curve and determine their nature.

A curve has equation y = 2x^(1/2) − 1. Find the equation of the normal to the curve at the point A (4, 3), giving your answer in the form y = mx + c.

The diagram shows part of the curve with equation y = x + 2/(2x − 1)². The lines x = 1 and x = 2 intersect the curve at P and Q respectively and R is the stationary point on the…

A curve has equation y = 3 tan((1/2)x) cos 2x. Find the gradient of the curve at the point for which x = (1/3)π. [5]

The curve with equation e^(2x) − 18x + y^3 + y = 11 has a stationary point at (p, q). Find the exact value of p. [4]

The diagram shows the curve with equation y = 6e^(−x/2). The points on the curve with x-coordinates 0 and 2 are denoted by A and B respectively. The shaded region is enclosed by…

The diagram shows the curve with parametric equations x = 3 ln(2t − 3), y = 4t ln t. The curve crosses the y-axis at the point A. At the point B, the gradient of the curve is 12.…

A curve has equation y = 3 tan(½x) cos 2x. Find the gradient of the curve at the point for which x = ⅓π.

The curve with equation e^(2x) − 18x + y³ + y = 11 has a stationary point at (p, q). Find the exact value of p.

Find the exact coordinates of the points on the curve y = x² / (1 − 3x) at which the gradient of the tangent is equal to 8.

The parametric equations of a curve are x = √t + 3, y = ln t, for t 0. Obtain a simplified expression for dy/dx in terms of t.

The diagram shows the curve y = x·e^(−(1/4)x²), for x ≥ 0, and its maximum point M. Find the exact coordinates of M.

The parametric equations of a curve are x = (ln t)², y = e^(2 − t²), for t 0. Find the gradient of the curve at the point where t = e, simplifying your answer.

The diagram shows the curve y = sin x cos 2x, for 0 ≤ x ≤ π, and a maximum point M, where x = a. The shaded region between the curve and the x-axis is denoted by R. Find the value…

Find the exact coordinates of the stationary points of the curve y = e^(3x²−1) / (1 − x²).

The equation of a curve is x³ + y² + 3x² + 3y = 4. Show that dy/dx = −(3x² + 6x) / (2y + 3).

The diagram shows the curve y = x cos 2x, for x ⩾ 0. Find the equation of the tangent to the curve at the point where x = ½π.