7. Differentiating Powers of a Function
by M. Bourne
Function of a Function
If y is a function of u, and u is a function of x, then we say
"y is a function of the function u".
Example 1
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In this section:
Chain Rule
Power Rule
Consider the function
y = (5x + 7)12.
If we let u = 5x + 7 (the inner-most expression), then we could write our original function as
y = u12
We have written y as a function of u, and in turn, u is a function of x.
This is a vital concept in differentiation, since many of the functions we meet from now on will be functions of functions, and we need to recognise them in order to differentiate them properly.
Chain Rule
To find the derivative of a function of a function, we need to use the Chain Rule:
`(dy)/(dx) = (dy)/(du) (du)/(dx)`
This means we need to
- Recognise `u` (always choose the inner-most expression, usually the part inside brackets, or under the square root sign).
- Then we need to re-express `y` in terms of `u`.
- Then we differentiate `y` (with respect to `u`), then we re-express everything in terms of `x`.
- The next step is to find `(du)/dx`.
- Then we multiply `dy/(du)` and `(du)/dx`.
Answer for Example 1
In Example 1,
y = u12, so `(dy)/(du) = 12u^11 = 12(5x+7)^11`
and
u = 5x + 7, so `(du)/(dx) = 5`
So
`dy/dx=(dy)/(du) (du)/(dx) ` `= (12(5x+7)^11)(5) ` `= 60(5x+7)^11`
Example 2
Find `dy/dx` if `y = (x^2+ 3)^5`.
Answer
In this case, we let u = x2 + 3 and then y = u5.
We see that:
- u is a function of x and
- y is a function of u.
For the chain rule, we firstly need to find `(dy)/(du)` and `(du)/(dx)`:
`(dy)/(du)=5u^4=5(x^2+3)^4`; and
`(du)/(dx)=2x`
So
`(dy)/(dx)=(dy)/(du)(du)/(dx)`
`=5(x^2+3)^4(2x)`
`=10x(x^2+3)^4`
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Play with the graph of this example on the Differentiation interactive applet page and explore what it means.
Example 3
Find `dy/dx` if `y=sqrt(4x^2-x)`.
Answer
In this case, we let u = 4x2 − x and then `y=sqrtu=u^(1/2)`.
Once again,
- u is a function of x and
- y is a function of u.
Using the chain rule, we firstly need to find:
`(dy)/(du)=1/2u^(-1/2)=1/(2sqrtu)=1/(2sqrt(4x^2-x)`
and
`(du)/(dx)=8x-1`
So
`(dy)/(dx)=(dy)/(du)(du)/(dx)`
`=1/(2sqrt(4x^2-x))(8x-1)`
`=(8x-1)/(2sqrt(4x^2-x))`
Note: How does `u^(-1//2)=1/sqrtu`? See:
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You can play with this example on the Differentiation interactive applet page.
The Derivative of a Power of a Function (Power Rule)
An extension of the chain rule is the Power Rule for differentiating. We are finding the derivative of un (a power of a function):
`d/dxu^n=n u^(n-1)(du)/dx`
Example 4
In the case of `y=(2x^3-1)^4` we have a power of a function.
Answer
If we let u = 2x3 - 1 then y = u4.
So now
- y is written as a power of u; and
- u is a function of x [ u = f(x) ].
To find the derivative of such an expression, we can use our new rule:
`d/(dx)u^n=n u^(n-1)(du)/(dx`
where u = 2x3 − 1 and n = 4.
So
`(dy)/(dx)=n u^(n-1)(du)/(dx)`
`=[4(2x^3-1)^3][6x^2]`
`=24x^2(2x^3-1)^3`
We could, of course, use the chain rule, as before:
`(dy)/(dx)=(dy)/(du)(du)/(dx`
Play with this example on the Differentiation interactive applet page.
Example 5
If `y=1/x`, find `dy/dx`
Answer
We can write `y=1/x` as `y=x^-1`. So:
`dy/dx=(-1)x^-2 = -1/x^2`
You can explore this example on the Differentiation interactive applet page.
CHALLENGE
Find the derivative of `y=(x^2(3x+1))/(x^4+2)`
Answer
In this example, we have a quotient, where the numerator is a product.
Once again, we let `y=u/v`, where `u=x^2(3x+1)` and `v=x^4+2`.
The quotient formula requires `(du)/(dx)` but u is a product.
Let `u = pq` where p = x2 and q = 3x + 1.
`(du)/(dx)=p(dq)/(dx)+q(dp)/(dx)`
`=(x^2)(3)+(3x+1)(2x)`
`=3x^2+6x^2+2x`
`=9x^2+2x`
We also require
`(dv)/(dx)=4x^3`
So
` (dy)/(dx)=(d(u/v))/(dx)=(v(du)/(dx)-u(dv)/(dx))/(v^2)`
`=((x^4+2)(9x^2+2x)-(x^2)(3x+1)(4x^3))/((x^4+2)^2)`
`=(9x^6+2x^5+18x^2+4x-12x^6-4x^5)/((x^4+2)^2)`
`=(-3x^6-2x^5+18x^2+4x)/((x^4+2)^2)`
Play with this challenge example on the Differentiation interactive applet page.
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