Mr. J Gallagher

Differentiation Worksheet

First Principles

Differentiate the following from first principles:

a) x2

b) 2x2 – 3x – 6

c) 3x2 + 7x

d) x2 + 6x

Tangent to a curve

Given Point on Tangent:

a) Find the equation of the tangent to the curve f (x) = 2x2 – 4x – 5 at the point (3,1).

b) Find the equation of the tangent to the curve f (x) = x2 – 6x at the point where x = 2.

c) Find the equation of the tangent to the curve f (x) = 2x2 – 4x – 5 at the point (3,1).

d) Find the equation of the tangent to the curve f (x) = 5 x2

1+ x2 at the point

(2,4).

Given Slope:

a) Find the coordinates of the point on the curve y = x2 + 3x – 1 at which the slope of the tangent to the curve is 5.

b) Find the coordinates of the point on the curve y = x2 + 4x + 6 at which the slope of the tangent to the curve is –2.

c) Let f (x) = 2 xx+2 , x ≠2, x ∈ R. Find the coordinates of the points at which

the slope of the tangent to the curve is 14 .

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Mr. J Gallagher

Given Point and Slope:

a) Find the value of k if the slope of the tangent to the curve y = x2 + kx is 3 at the point where x = -1.

b) A curve is defined by the equation y = ax2 + b, where a, b are constants. If the slope of the curve at the point (2, -2) is 3. Find the values of a and b.

c) Given that the curve with equation y = ax2 + bx + 5 has slope 4 at the point (5,0). Find the value of the constants a and b.

Maximum / Minimum

For each of the following functions:

Calculate the maximum and minimum values.

Find the point(s) where the curve crosses the y-axis.

Hence draw a sketch of the curve.

Give the range of values for which the curve is increasing / decreasing.

Calculate the point of inflection.

State whether the functions are Injective, Surjective or Bijective. Give a reason for your answer.

a) f (x) = x3 – 9x2 + 24x – 20.

b) f (x) = x3 – 6x2 + 9x + 2

c) f (x) = x3 + 3x2 + 1

d)

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Mr. J Gallagher

Real Life Examples

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