Differentiate the following:
Differentiate the following:
Differentiate y = \dfrac{2}{\sqrt{x}}. Express your answer in surd form.
Consider the function y = \dfrac{7}{x}.
Rewrite the function in negative index form.
Find the derivative, expressing your answer with a positive index.
Differentiate y = \dfrac{1}{4 x^{3}}. Express your answer in positive index form.
Consider the function y = \dfrac{5 x \sqrt{x}}{4 x^{5}}.
Rewrite the function in simplified negative index form.
Find \dfrac{dy}{dx}.
Differentiate y = \dfrac{2}{x^{a}} - \dfrac{3}{x^{b}}, where a and b are constants.
Consider the function f \left( r \right) = \dfrac{2}{r} + \dfrac{r}{3}.
Rewrite the function so that each term is a power of r.
Find f' \left( r \right).
Consider the function y = \dfrac{8 x^{2} + 6 x + 4}{\sqrt{x}}.
Rewrite the function so that each term is a power of x.
Hence, find the derivative of the function.
For each of the following:
Express the function in expanded form.
Find the derivative of the function.
y = \left( 6 x + 5\right) \left(x + 3\right)
y = 2 x^{2} \left( 7 x + 2\right)
y = \left(x + 4\right)^{2}
y = \dfrac{4}{9} \left( - 4 x - 8\right)
Consider the function f \left( x \right) = \left(\sqrt{x} + 10 x^{2}\right)^{2}
Rewrite the function f \left( x \right) in expanded form, with all terms written as powers of x.
Hence, differentiate the function.
For each of the following:
Rewrite the function in expanded form.
Hence, find the derivative.
y = \left(\dfrac{4}{x} + 2 \sqrt{x}\right) \left(5 + \dfrac{6}{\sqrt{x}}\right)
y = \left(\sqrt[5]{x} + 2 \sqrt{x}\right)^{2}
y = \left( 3 \sqrt{x} + \dfrac{2}{x}\right)^{2}
y = \left( 4 \sqrt{x} - \dfrac{1}{\sqrt{x}}\right) \left( 4 \sqrt{x^{3}} + \dfrac{1}{x}\right)
y = \left( 2 x + \dfrac{3}{x}\right) \left( 6 \sqrt{x} + 5\right)
The tangent to the curve y = 3 + \dfrac{x}{x + 2} at the point \left(0, 3\right) has the equation \\ y = \dfrac{1}{2} x + 3:
Find f' \left( 0 \right).
Each of the following graphs contain a curve, f \left( x \right), along with one of its tangents, g \left( x \right).
State the coordinates of the point at which g \left( x \right) is a tangent to the curve f \left( x \right).
State the gradient of the tangent.
Hence, determine the equation of the line y = g \left( x \right).
Consider the curve f \left( x \right) drawn along with g \left( x \right), which is a tangent to the curve:
What are the coordinates of the point at which g \left( x \right) is a tangent to the curve f \left( x \right)?
What is the gradient of the tangent line?
Hence, determine the equation of the line y = g \left( x \right).
What is the x-coordinate of the point on the curve at which we could draw a tangent line that has the same gradient as g \left( x \right)?
Consider the tangent to the curve f \left( x \right) = x^{2} at the point \left( - 1 , 1\right).
Find the gradient of the function f \left( x \right) = x^{2} at this point.
Find the equation of the tangent to the curve f \left( x \right) = x^{2} at the point \left( - 1 , 1\right).
Consider the tangent to the curve f \left( x \right) = x^{3} at the point \left( - 1 , -1\right).
Find the gradient of the function f \left( x \right) = x^{3} at this point.
Hence find the equation of the tangent to the curve f \left( x \right) = x^{3} at the point \left( - 1 , -1\right).
Consider the tangent to the curve f \left( x \right) = - x^{2} at the point \left( - 3 , - 9 \right).
Find the gradient of the function f \left( x \right) = - x^{2} at this point.
Hence, find the equation of the tangent to the curve f \left( x \right) = - x^{2} at the point \left( - 3 , - 9 \right).
Consider the tangent to the curve f \left( x \right) = - x^{3} at the point \left(2, - 8 \right).
Find the gradient of the function f \left( x \right) = - x^{3} at this point.
Hence, find the equation of the tangent to the curve f \left( x \right) = - x^{3} at the point \left(2, - 8 \right).
Consider the tangent to the curve f \left( x \right) = 3 x^{3} at x = 2.
Describe what must be done to find the equation of the tangent to the curve f(x) at \\ x=2.
Hence, find the equation of the tangent to the curve f \left( x \right) = 3 x^{3} at x = 2.
Consider the tangent to the curve f \left( x \right) = 3 x^{3} at x = 2.
State what two pieces of information we need in order to find the equation of the tangent line at x = 2.
Hence, determine the equation of the tangent to the curve f \left( x \right) = 3 x^{3} at x = 2.
Consider the tangent to the curve f \left( x \right) = 6 \sqrt{x} at the point \left(4, 12\right).
Find the gradient of the function f \left( x \right) = 6 \sqrt{x} at this point.
Hence, find the equation of the tangent to the curve f \left( x \right) = 6 \sqrt{x} at the point \left(4, 12\right).
Consider the tangent to the curve f \left( x \right) = 5 \sqrt{x} at the point \left(\dfrac{1}{9}, \dfrac{5}{3}\right).
Find the gradient of the function f \left( x \right) = 5 \sqrt{x} at this point.
Hence, find the equation of the tangent to the curve f \left( x \right) = 5 \sqrt{x} at the point \left(\dfrac{1}{9}, \dfrac{5}{3}\right).
Consider the tangent to the curve f \left( x \right) = - 2 x^{2} + 8 x + 2 at x = 1.
Find the y-coordinate of the point of intersection between the tangent and the curve.
Hence determine the equation of the tangent to the curve f \left( x \right) = - 2 x^{2} + 8 x + 2 at \\ x = 1.
Find the equation of the tangent to the curve f \left( x \right) = \dfrac{2}{x^{3}} at x = - 2.
Find the equation of the tangent to the curve f \left( x \right) = 0.3 x^{3} - 5 x^{2} - x + 4 at x = 1.
Find the equation of the tangent to the curve f \left( x \right) = \dfrac{9 x + 4}{3 x} at x = - 1.
By considering the graph of f \left( x \right) = 2 x, find f'\left( - 5 \right).
Find the gradient of f \left( x \right) = x^{5} - 3 x^{4} at the point \left(3, 0\right).
Find the gradient of f \left( x \right) = \dfrac{6}{\sqrt{x}} at the point \left(25, \dfrac{6}{5}\right).
Consider the function f \left( x \right) = 6 x^{2} + 5 x + 2.
Find f' \left( x \right).
Find f' \left( 2 \right).
Find the x-coordinate of the point at which f' \left( x \right) = 41.
Consider the parabola f \left( x \right) = x^{2} + 3 x - 10.
Find the x-intercepts.
Find the gradient of the tangent at the positive x-intercept.
Consider the curve given by the function f \left( x \right) = x^{3} + 5 x.
Find the gradient of the tangent at the point \left(2, 18\right).
Consider the function y = 4 x^{2} - 5 x + 2. Find the value of x at which the tangent to the parabola is parallel to the x-axis.
Consider the function y = x^{2} - 3 x + 4. Find the x-coordinate of the point on the curve where the tangent makes an angle of 45 \degree with the x-axis.
Consider the function f \left( x \right) = 5 x^{2} + \dfrac{4}{x} - 1. The tangent to the curve at the point \left(2, 21\right) makes an angle of \theta with the x-axis. Find \theta, correct to the nearest degree.
Consider the function f \left( x \right) = x^{2} + 5 x.
Find the x-coordinate of the point at which f \left( x \right) has a gradient of 13.
Hence, state the coordinates of the point on the curve where the gradient is 13.
Consider the function f \left( x \right) = x^{3} + 3 x^{2} - 19 x + 2.
Find the x-coordinates of the points at which f \left( x \right) has a gradient of 5.
Hence, state the coordinates of the points on the curve where the gradient is 5.
Consider the function f \left( x \right) = 2 x^{2} - 216 \sqrt{x}. Find the coordinates of the point on the curve where the gradient is 0.
Find the x-coordinate(s) of the point(s) at which f \left( x \right) = \left(x - 5\right) \left(x^{2} + 3\right) has a gradient of 0.
Consider the function f \left( x \right) = x^{3} + 6 x^{2} - 14 x - 2.
Find the x-coordinate(s) of the point(s) on the curve where the gradient is the same as that of g \left( x \right) = 22 x - 4.
Hence, state the coordinates of the points on the curve where the gradient is the same as that of g \left( x \right) = 22 x - 4.
At point M, the equation of the tangent to the curve y = x^{2} is given by y = 4 x - 4. Find the coordinates of M.
At point M, the equation of the tangent to the curve y = x^{3} is given by y = 12 x - 16. Find the coordinates of M.
Consider the function f \left( x \right) = \dfrac{4 x^{3}}{3} + \dfrac{5 x^{2}}{2} - 3 x + 7. Find the x-coordinates of the points on the curve whose tangent is parallel to the line y = 3 x + 7.
5 x + y + 2 = 0 is the tangent line to the curve y = x^{2} + b x + c at the point \left(9, - 47 \right).
Find the derivative \dfrac{d y}{d x} of y = x^{2} + b x + c.
State the gradient of the tangent to the curve at x = 9.
Solve for the value of b.
Solve for the value of c.
The curve f \left( x \right) = k \sqrt{x} - 5 x has a gradient of 0 at x = 16. Find the value of k.
In the following graph, the line y = \dfrac{1}{10} x + b is tangent to the graph of f \left( x \right) = 6 \sqrt{x} at \\ x = a.
Find the values of a and b.
The curve y = a x^{3} + b x^{2} + 2 x - 17 has a gradient of 58 at the point \left(2, 31\right). Find the values of a and b.
The graph of y = a x^{3} + b x^{2} + c x + d intersects the x-axis at \left(2, 0\right), where it has a gradient of 36. It also intersects the y-axis at y = - 28, where the tangent is parallel to the x-axis.
Find the values of a, b, c and d.
Consider the curve f \left( x \right) = x^{2} + 8 x + 15 at the point \left(4, 63\right).
Find f' \left( x \right).
Find the gradient of the tangent to the curve at the point.
Find the equation of the tangent to the curve at the point.
Find the gradient of the normal to the curve at the point.
Find the equation of the normal to the curve at the point.
Consider the curve f \left( x \right) = 4 x + \dfrac{64}{x} at the point \left(4, 32\right).
Find the equation of the tangent to the curve at the point.
Find the equation of the normal to the curve at the point.
The normal to the curve f \left( x \right) = x^{2} + 3 at the point A \left(5, 28\right) meets the curve again at B.
Find the equation of the normal to the curve at point A.
Solve for the x-coordinate of point B.
- 4 x + y + 1 = 0 is the normal to the curve y = x^{2} + b x + c at the point \left( - 8 , - 32 \right).
Find the derivative \dfrac{d y}{d x} of y = x^{2} + b x + c.
State the gradient of the normal to the curve at x = - 8.
Find the value of b.
Find the value of c.