Differentiate
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Differentiate
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Write down the formula that should be used as a starting point when explaining differentiation from first principles.
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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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A student is trying to show that the derivative of is using first principles.
Their working is shown below.
Find and explain their error.
STEP 1
STEP 2
STEP 3
STEP 4
STEP 5
STEP 6
When h=0,
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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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Prove, from first principles, that the derivative of 4x is 4.
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Prove, from first principles, that the derivative of -3x is -3.
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Prove, from first principles, that the derivative of 2x2 is 4x.
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For each of the following, find in terms of x:
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Given that , , find .
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For each of the following, find in terms of x:
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The function f 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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The function f is defined by
Find .
Given that the equation has exactly one real solution, find the value of c.
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A curve is described by the equation .
Make y the subject of the equation.
Hence find .
Find the coordinates of the point on the curve where the gradient is -2.
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The curve with equation has a gradient of -7 at the point , and a gradient of -3 at the point (1, 3).
By considering show that and .
Hence find the values of a and b.
By considering a point that you know to be on the curve, find the value of c.
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Prove, from first principles, that the derivative of is 2ax, where a is a constant.
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Prove, from first principles, that the derivative of is .
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For each of the following, find in terms of x:
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Given that , find .
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For each of the following, find in terms of :
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The function f is defined by . Show that there are no solutions to the equation .
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A curve has the equation .
Show that , where a and b 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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A curve has the equation , where b is a constant. Given that there is only one point on the curve where the gradient is zero, determine the possible values of b.
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A curve is described by the equation .
By rearranging the equation to make y the subject, find .
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The curve with equation has a gradient of 8 at the point , and a gradient of -10 at the point . Find the values of and c.
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Prove, from first principles, that the derivative of is .
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Show that
Prove, from first principles, that the derivative of is .
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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 x:
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The function f is defined by. Determine the range of values for p for which the equation has at least one real solution.
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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 f is defined by . Determine the relationship between the value of n 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 (-1, 4). At the point (2, 7) the gradient of the curve is 7. Find the values of a, b and c.
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