I have been asked to solve this two different ways. The first way is to use the chain rule and then simplify (which I have already done properly), and the second way is to simplify and then differentiate (not necessarily with the chain rule). I am stuck on how to simplify this question and then how to differentiate it.
Square Root(9x^2) ??
Also I am stuck on how to use the chain rule to differentiate and then simplify this question, as well as the second way where I have to simplify and then differentiate it:
(8x^3)^2/3
I would appreciate if someone would be able to explain these two questions to me, thank you.
y = [ 9 x^2 ] ^.5 ??
let z = 9 x^2 then dz/dx = 18 x
then we have y = z^.5
dy/dz = .5 z^-.5 = .5/3x
but chain rule
dy/dx = dy/dz*dz/dx (.5 /3x)(18 x)
= 9/3 = 3 at last
simplify first
y = (9 x^2)^.5 = 3x
dy/dx = 3 the end
(8x^3)^2/3
I am not going to struggle through that with the chain rule. Go about it the same way as I did upstairs, z = 8 x^3
simplify method:
remember (8x^3)^(1/3) = 2x
(2x)^2 = 4 x^2
y = 4x^2
dy/dx = 8 x the end
To solve the first problem, which is to simplify and then differentiate the expression square root of 9x^2, we can first simplify the expression and then differentiate it.
1. Simplify:
The square root of 9x^2 can be written as √(9x^2). Since 9 is a perfect square (3*3 = 9), we can take the square root of 9 to simplify the expression:
√(9x^2) = 3x.
2. Differentiate:
Now that we have simplified the expression to 3x, we can differentiate it with respect to x. Since 3x has a power of 1, the derivative will be the coefficient of x, which is 3:
d/dx (3x) = 3.
So, the derivative of the square root of 9x^2 is 3.
For the second problem, which is to use the chain rule to differentiate and then simplify the expression (8x^3)^(2/3), we need to apply the chain rule first and then simplify the expression.
1. Apply the chain rule:
To differentiate this expression using the chain rule, we will differentiate the outer function and multiply it by the derivative of the inner function.
The outer function is ( )^(2/3), and the inner function is 8x^3.
The derivative of the outer function is (2/3) * ( )^((2/3) - 1), and the derivative of the inner function is 24x^2.
2. Simplify:
Now that we have the derivatives of the outer and inner functions, we can simplify the expression by multiplying them together:
[(2/3) * ( )^((2/3) - 1)] * 24x^2.
Simplifying the derivative expression:
[(2/3) * (8x^3)^((2/3) - 1)] * 24x^2.
In the exponent, (2/3) - 1 can be simplified to (2/3) - (3/3) = (2/3) - (1/3) = 1/3.
So, the expression becomes:
[(2/3) * (8x^3)^(1/3)] * 24x^2.
Further simplification:
[(2/3) * 2x] * 24x^2.
(4/3) * 24x^3.
32x^3.
So, after applying the chain rule and simplifying, the derivative of (8x^3)^(2/3) is 32x^3.