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) = 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) = 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) = 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.