Trying to solve for the derivative

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In summary, this conversation is a helpful reminder for Homework Statement Hi, I'm trying to find the derivative of a specific equation and I feel like I messed up somewhere but can't figure out exactly what.Homework Equations##y = \frac{x^8}{8(1-x^2)^4}##The Attempt at a SolutionFirst I calculated the derivatives for the denominator and numerator:##\frac{\mathrm d}{\mathrm d x} \big( x^8 \big) = 8x^7####\frac{\mathrm d}{\mathrm d x} \big( 8(1-x^2
  • #1
Mootjeuh
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Homework Statement


Hi, I'm trying to find the derivative of a specific equation and I feel like I messed up somewhere but can't figure out exactly what.

Homework Equations


##y = \frac{x^8}{8(1-x^2)^4}##

The Attempt at a Solution


First I calculated the derivatives for the denominator and numerator:
##\frac{\mathrm d}{\mathrm d x} \big( x^8 \big) = 8x^7##

##\frac{\mathrm d}{\mathrm d x} \big( 8(1-x^2)^4 \big) = 32(1-x^2)^3(-2x)##

Then the actual attempt at solving:
##\frac{\mathrm d}{\mathrm d x} \big( \frac{x^8}{8(1-x^2)^4} \big) = \frac{8(1-x^2)^48x^7-x^832(1-x^2)^3(-2x)}{[8(1-x^2)^4]^2}##

##\frac{8(1-x^8)8x^7-x^832(1-x^6)(-2x)}{[8(1-x^8)]^2} = \frac{8(1-x^8)8x^7-x^832(1-x^6)(-2x)}{(8-8x^8)^2 = 64-128x^8+64x^{16}}##

##\frac{8(1-x^8)8x^7-x^832(1-x^6)(-2x)}{64-128x^8+64x^{16}}##
 
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  • #2
##(1-x^2)^4## is not equal to ##1-x^8##
 
  • #3
Mootjeuh said:

Homework Statement


Hi, I'm trying to find the derivative of a specific equation and I feel like I messed up somewhere but can't figure out exactly what.

Homework Equations


##y = \frac{x^8}{8(1-x^2)^4}##

The Attempt at a Solution


First I calculated the derivatives for the denominator and numerator:
##\frac{\mathrm d}{\mathrm d x} \big( x^8 \big) = 8x^7##

##\frac{\mathrm d}{\mathrm d x} \big( 8(1-x^2)^4 \big) = 32(1-x^2)^3(-2x)##

Then the actual attempt at solving:
##\frac{\mathrm d}{\mathrm d x} \big( \frac{x^8}{8(1-x^2)^4} \big) = \frac{8(1-x^2)^48x^7-x^832(1-x^2)^3(-2x)}{[8(1-x^2)^4]^2}##

##\frac{8(1-x^8)8x^7-x^832(1-x^6)(-2x)}{[8(1-x^8)]^2} = \frac{8(1-x^8)8x^7-x^832(1-x^6)(-2x)}{(8-8x^8)^2 = 64-128x^8+64x^{16}}##

##\frac{8(1-x^8)8x^7-x^832(1-x^6)(-2x)}{64-128x^8+64x^{16}}##

imiuru said:
##(1-x^2)^4## is not equal to ##1-x^8##

Nor is (1 - x2)3 = 1 - x6

Your work to here looks fine:
$$\frac{8(1-x^2)^48x^7-x^832(1-x^2)^3(-2x)}{[8(1-x^2)^4]^2}$$
Expanding the (1 - x2) factors as you did is not helpful, even if you had done them correctly. Instead, find the greatest common factor of the two terms in the numerator, and factor it out of both terms. Notice that you have (1 - x2) to some power in both terms, and you have x to some power in both terms. There is also a coefficient of 64 in the first term and one of -64 in the second term.
 
  • #4
Mootjeuh,
Based on the mistakes in this thread and in the other thread you posted about logs, since you are apparently studying calculus, it would be very useful for you to spend some time reviewing algebra concepts from your previous classes. If you don't have a good working knowledge of how to manipulate algebra expressions, you will have a very difficult time following explanations in your calculus class.
 

1. What is a derivative?

A derivative is a mathematical concept that represents the rate of change of a function at a specific point. It tells us how much a function is changing at a particular point on its graph.

2. Why do we need to solve for the derivative?

Solving for the derivative allows us to find the instantaneous rate of change of a function at a specific point. This is useful in many areas of mathematics and science, such as calculating velocity, acceleration, and optimization problems.

3. How do you solve for the derivative?

To solve for the derivative, we use a process called differentiation. This involves applying specific rules and formulas to a given function to find its derivative. These rules include the power rule, product rule, quotient rule, and chain rule.

4. What are some applications of finding derivatives?

Finding derivatives has many practical applications in fields such as physics, engineering, economics, and statistics. It can be used to calculate the velocity and acceleration of an object, determine the optimum production level in economics, and analyze data trends in statistics.

5. Can you give an example of solving for the derivative?

Sure, for example, if we have the function f(x) = 2x^2 + 3x, we can use the power rule to find its derivative. The power rule states that the derivative of x^n is nx^(n-1). So, the derivative of f(x) would be f'(x) = 4x + 3. This tells us that the slope of the tangent line to the graph of f(x) is 4x + 3 at any given point on the graph.

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