1. Differentiating Polynomials & Equations of Tangents & Normals
Introduction to Calculus 2:
Differentiating Polynomials & Equations of Tangents & Normals
Independent Practice:
Oxford Text:
Exercises p205, 7F; p207, 7G
Pearson Chapter 11 p380,

2. More Rules for Polynomials
The derivative of any constant is zero.
The graph of 𝑓 𝑥 =𝑐 (where c is any real number) is a horizontal line which has a gradient of zero.
The derivative of a constant times a function is the constant times the derivative of a function.
If 𝑦=𝑐𝑓 𝑥 (where c is any real number) then 𝑦 ′ =𝑐𝑓′(𝑥)
Example: Find the derivative of 𝑓 𝑥 =3 𝑥 using first principles.

3. More Rules for Polynomials
The derivative of a function that is the sum or difference of two or more terms is the sum or difference of the derivatives of the terms.
If 𝑓 𝑥 =𝑢(𝑥)±𝑣(𝑥) then 𝑓 ′ 𝑥 =𝑢′(𝑥)±𝑣′(𝑥)
Differentiate the following functions:
𝑦= 𝑥 − 5 𝑥 3 +𝑥
𝑦=𝑥 𝑥 − 2𝑥
𝑦=2 𝑥 3 +5𝑥−9

4. More Examples c) 𝑦= ( 𝑥 − 2 𝑥 ) 2 𝑎) 𝑦= 𝑥 −9, x ≥ 0 b) 𝑦= 𝑥 3 +2 2
Differentiate the following functions:
c) 𝑦= ( 𝑥 − 2 𝑥 ) 2
𝑎) 𝑦= 𝑥 −9, x ≥ 0
b) 𝑦= 𝑥
d) 𝑦= 5 𝑥 2 +4𝑥−3 𝑥
f) 𝑦= 2 𝑥 4 −4𝑥+3 𝑥
e) 𝑦= 𝑥 (7 𝑥 2 −3𝑥+2)

5. Equations of Tangent and Normal Lines
The tangent is a straight line that touches the curve at one point. The slope of the tangent is the derivative of the function at that point.
The normal line at a point on a curve is the straight line perpendicular to the tangent at that point.
𝑚 𝑁 =− 1 𝑚 𝑇
*Remember:
To write the equation of a line you must know a slope and the coordinate of a point
the y-value of the point on the curve can be found by substituting the x-value into the function f(x).
the tangent slope (and consequently the normal slope) can be found by substituting the x-value into the derivative function f’(x)
use the “point-slope” form of the equation of a line to write the equation of the line (usually the easiest form)

6. Examples
1. a) Write the equation of the tangent to 𝑔 𝑥 = 𝑥 2 𝑥−1 at the point x = 2
b) Write the equation of the normal to 𝑔 𝑥 = 𝑥 2 𝑥−1 at the point x = 2

7. Examples - Continued
2. a) Write the equation of the tangent to f 𝑥 =1−3𝑥+12 𝑥 2 −8 𝑥 3 at the point (1, 2).
b) Write the equation of the tangent to f 𝑥 =1−3𝑥+12 𝑥 2 −8 𝑥 3 which is parallel to the tangent at (1, 2).

8. Examples - Continued
3. Find k if the tangent to f 𝑥 =2 𝑥 3 +𝑘 𝑥 2 −3 at the point where x = 2 has a gradient = 4.
4. Find the equation of the normal to f 𝑥 = 5 𝑥 − 𝑥 at the point (1, 4)