Prove Power Rule: Implicit Differentiation w/ Rational Exponents

In summary, The conversation involves someone asking for help with proving the power rule for a rational number using a specific method. The method involves writing the equation in a specific form and differentiating implicitly. The person also mentions the stipulation of assuming the power rule holds for integers and asks for help with where they are stuck. They are reminded not to post multiple times.
  • #1
mattxr250
8
0
Ok guys, I'm new here and I need some help with a math problem...

The problem asks me to prove the power rule ---> d/dx[x^n] = nx^(n-1) for the case in which n is a ratioinal number...

the one stipulation is that I have to prove it using this method: write y=x^(p/q) in the form y^q = x^p and differentiate implicitly...assume that p and q are integers, where q>0.

Thanks for any help

Matt
 
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  • #2
Would I be correct in guessing that we are allowed to assume that the power rule holds for integers? If so, then it is simply a matter of differentiating both sides of the equation [tex]y^q=x^{p}[/tex] and then solving for y'. Where are you stuck?
 
  • #3
Don't multiple post.
 

1. What is the Power Rule in calculus?

The Power Rule in calculus is a fundamental rule that allows us to find the derivative of a power function. It states that if we have a function of the form f(x) = x^n, then its derivative is given by f'(x) = nx^(n-1). In other words, we can bring the exponent down in front of the variable and subtract 1 from the original exponent.

2. What is implicit differentiation?

Implicit differentiation is a technique used to find the derivative of a function that is not explicitly defined in terms of x. This means that the function may contain both x and y terms, making it difficult to use the traditional methods of finding derivatives. To use implicit differentiation, we treat y as a function of x and use the chain rule to find the derivative of the function.

3. How do you apply implicit differentiation to functions with rational exponents?

To apply implicit differentiation to functions with rational exponents, we first rewrite the function in exponential form. Then, we use the power rule for derivatives to find the derivative of the function. Finally, we substitute the original expression for y back into the derivative to find the final result.

4. What are the steps for proving the Power Rule using implicit differentiation?

The steps for proving the Power Rule using implicit differentiation are as follows:
1. Rewrite the function in exponential form.
2. Apply the power rule for derivatives to find the derivative of the function.
3. Substitute the original expression for y back into the derivative.
4. Simplify the expression to get the final result, which should match the Power Rule.

5. Can the Power Rule be extended to functions with irrational exponents?

Yes, the Power Rule can be extended to functions with irrational exponents. The general rule is that if we have a function of the form f(x) = x^n, where n is any real number, the derivative is given by f'(x) = nx^(n-1). This means that the Power Rule can be applied to functions with both rational and irrational exponents, as long as the exponent is a real number.

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