Differentiate
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Differentiate
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By first defining an appropriate function , where is a function of , show that the function defined by
may be written in the form
where is a positive integer.
Using your result from part (a) along with the chain rule
find for the function defined in part (a). Be sure to give your answer entirely in terms of .
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Write down the gradient of the line with equation , where k is a constant.
Find the gradient at the point where for the following functions
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Given that
By first defining an appropriate function and writing in the form where is a positive integer, use the chain rule to find in terms of x.
determine the nature of those stationary points.
Sketch the curve of .
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Given that , find .
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Find the x-coordinate of the point on the curve where the gradient is 4.
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Find the coordinates of the points on the curve where the gradient is 0.
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Find when
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The function is given by
Show that can be written in the form , where a,b, c and d are constants to be found.
Find .
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For each of the following, find in terms of :
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Given that find .
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For each of the following, find in terms of :
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By first defining an appropriate function , where is a function of , show that the function defined by
may be written as a function of .
Using your result from part (a) along with the chain rule
find for the function defined in part (a). Be sure to give your answer entirely in terms of .
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The function is defined by .
Find
Solve the equation
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A curve has the equation
Find
Find the coordinates of the point on the curve where the gradient is 2.
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Given that
use the chain rule to find
find the coordinates of any stationary points and determine their nature
sketch the curve.
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The function is defined by
Find
Given that the equation has exactly one real solution, find the value of .
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A curve is described by the equation
Make the subject of the equation.
Hence find .
Find the coordinates of the point on the curve where the gradient is .
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The curve with equation has a gradient of at the point , and a gradient of at the point
By considering show that and .
Hence find the values of and .
By considering a point that you know to be on the curve, find the value of .
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For each of the following, find in terms of :
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Given that
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For each of the following, find in terms of :
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For each of the following, use the chain rule to find in terms of :
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The function is defined by Show that there are no solutions to the equation
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A curve has the equation
Show that where and are rational numbers to be found.
Hence find the coordinates of the point on the curve where the gradient is 0.
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Given that
use the chain rule to find
find the coordinates of any stationary points and determine their nature
sketch the curve.
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A curve has the equation , where is a constant. Given that there is only one point on the curve where the gradient is zero, determine the possible values of .
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A curve is described by the equation
By rearranging the equation to make the subject, find .
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The curve with equation has a gradient of at the point and a gradient of at the point . Find the values of , and .
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For each of the following, find in terms of :
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Given that
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For each of the following, find in terms of :
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For each of the following, use the chain rule to find in terms of :
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A curve has the equation Find the coordinates of the point on the curve where the gradient is 0.
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The function is defined by Determine the range of values for for which the equation has at least one real solution.
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Given that
use the chain rule to find
find the coordinates of any stationary points and determine their nature
sketch the curve.
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The function is defined by Determine the relationship between the value of and the number of real solutions to the equation
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A curve is described by the equation Find .
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The curve with equation passes through the point At the point the gradient of the curve is 7. Find the values of and .
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