Page 1
Differentiating xn
This is the one of the most frequently used rules of differentiation.
If y=xn thendydx=nxn−1
This can be used to differentiate surds or fractions involving powers of
x .
For example,
√x can also be written as
x12 , allowing it to be easily differentiated using the above rule.
3x can also be written as
3x−1 , so
ddx(3x)=−3x−2
Page 2
Differentiating functions
ddx is a linear operator. This means it can be separated out over addition or subtraction and constants can be moved outside the operation. Therefore, ddx(xm+xn)=ddx(xm)+ddx(xn)
and
ddx(kxm)=kddx(xm)
where
k is any constant.
Page 3
Example 1
Example 1: Differentiate the following function: y=3√x+4x2
Answer: As demonstrated on the previous page, we can treat the two terms in this function as if they were separate functions and their constants do not change with differentiation.
First, rewrite the function using exponents instead of surds and fractions:
y=3x12+4x−2
and then differentiatie:
dydx=3(12)x12−1+4(−2)x−2−1=32x−12−8x−3
This can be simplified to give:
dydx=32√x−8x−3
Page 4
Example 2
Example 2: Differentiate the following function: y=x2−3x−4x+1
Answer: Where there's a fraction with
x terms on top and bottom, factorise the top before continuing.
This gives:
y=(x−4)(x+1)x+1
Clearly
(x+1) can be factored from top and bottom to give
y=x−4
which is the same as
y=x1−4x0
and then differentiate:
dydx=1