Differentiating twice with respect to x

In summary: The question asks for the second order derivative of z, so I think the y is a typo and should be z.In summary, the problem is to differentiate twice the given function z = sinx using the product and chain rules. The initial attempt uses the product rule, but the book's answer incorporates the chain rule as well. The confusion may be due to a typo, as the question is part of a larger problem involving a substitution of z = sinx.
  • #1
sooty1892
5
0

Homework Statement



Differentiate twice: z = sinx


Homework Equations



Product rule
Chain rule

The Attempt at a Solution



dy/dx = dy/dz * dz/dx

dy/dx = dy/dz * cosx

Using the product rule:

d^2y/dx^2 = d^2y/dz^2 * cosx - dy/dz * sinx

According to the answer in the book the answer is: d^2y/dx^2 = d^2y/dz^2 * cos^2x - dy/dz * sinx but I don't see how.

Thanks for any help.
 
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  • #2
Why are you considering the product and chain? Product is used when you find the derivative of a product f.g , and chain is used when you have an expression f(g(x)),
and I don't see how you have either of these.
 
  • #3
Ok. You're right that I don't use the chain rule but the product rule is used when differentiating dy/dx = dy/dz * cosx since there's 2 parts which both have variables.
 
  • #4
sooty1892 said:

Homework Statement



Differentiate twice: z = sinx


Homework Equations



Product rule
Chain rule

The Attempt at a Solution



dy/dx = dy/dz * dz/dx

dy/dx = dy/dz * cosx

Using the product rule:

d^2y/dx^2 = d^2y/dz^2 * cosx - dy/dz * sinx

According to the answer in the book the answer is: d^2y/dx^2 = d^2y/dz^2 * cos^2x - dy/dz * sinx but I don't see how.

Thanks for any help.

You need to be more careful when posing questions. The question you wrote has no y in it anywhere. Did you mean "find d^y/dx^2, when y = f(z) and z = sin(x)"?

RGV
 
  • #5
I think I do. This is part of a much larger second order differential equations question where z = sin(x) is a substitution.
 

FAQ: Differentiating twice with respect to x

1. What does it mean to differentiate twice with respect to x?

Differentiating twice with respect to x means taking the derivative of a function with respect to x, and then taking the derivative of that derivative with respect to x. This results in a second derivative, which represents the rate of change of the slope of the original function.

2. Why is it important to differentiate twice with respect to x?

Differentiating twice with respect to x can provide valuable information about the behavior and characteristics of a function. It can help determine the concavity of a curve, identify maximum and minimum points, and calculate the rate of change of the slope at a specific point.

3. What is the difference between first and second derivatives?

The first derivative represents the instantaneous rate of change of a function at a specific point, while the second derivative represents the rate of change of the slope of the function. In other words, the first derivative tells us how fast the function is changing, while the second derivative tells us how fast the slope of the function is changing.

4. How do you find the second derivative using the power rule?

The power rule states that to find the derivative of a function raised to a power, you multiply the power by the coefficient, subtract 1 from the power, and then take the original function to the power minus 1. To find the second derivative using the power rule, you simply apply this rule twice, taking the derivative of the first derivative.

5. Are there any other methods for finding the second derivative?

Yes, there are other methods for finding the second derivative, such as using the product rule, quotient rule, or chain rule. These methods are useful for finding the second derivative of more complex functions that cannot be easily solved using the power rule.

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