Compute the Derivative of A with Respect to x

Oct 13, 2025 by talynt.org 113 views

Compute the Derivative of A with Respect to x

In this article, we will explore the concept of derivatives and how to compute them. Derivatives are a fundamental concept in calculus and are used to measure the rate of change of a function with respect to one of its variables. In this case, we will compute the derivative of A with respect to x from the given equation.

The equation given is:

${ A = \frac{26.5}{x^{0.45}} }$

This equation represents a function A that depends on the variable x. To compute the derivative of A with respect to x, we will use the power rule of differentiation.

The power rule of differentiation states that if we have a function of the form:

${ f(x) = x^n }$

then the derivative of f with respect to x is:

${ f'(x) = nx^{n-1} }$

This rule can be extended to functions of the form:

${ f(x) = x^a }$

where a is a constant. In this case, we have:

${ A = \frac{26.5}{x^{0.45}} }$

which can be rewritten as:

${ A = 26.5x^{-0.45} }$

Now that we have rewritten the equation in the form of the power rule, we can apply the rule to compute the derivative of A with respect to x.

${ \frac{dA}{dx} = -0.45 \cdot 26.5x^{-0.45-1} }$

${ \frac{dA}{dx} = -11.775x^{-1.45} }$

We can simplify the derivative by rewriting it in a more convenient form.

${ \frac{dA}{dx} = -\frac{11.775}{x^{1.45}} }$

Now that we have the derivative, we can compute its value at x = 0.23.

${ \frac{dA}{dx} = -\frac{11.775}{(0.23)^{1.45}} }$

${ \frac{dA}{dx} = -\frac{11.775}{0.2343} }$

${ \frac{dA}{dx} = -50.23 }$

We are asked to round our answer to two decimal places. Therefore, we round -50.23 to -50.23.

In this article, we computed the derivative of A with respect to x from the given equation. We applied the power rule of differentiation and simplified the derivative to obtain a more convenient form. Finally, we computed the value of the derivative at x = 0.23 and rounded our answer to two decimal places.

The final answer is: -50.23
Derivative of A with Respect to x: Q&A

In our previous article, we computed the derivative of A with respect to x from the given equation. In this article, we will answer some frequently asked questions related to the derivative of A with respect to x.

A: The derivative of A with respect to x is:

${ \frac{dA}{dx} = -\frac{11.775}{x^{1.45}} }$

A: To compute the derivative of A with respect to x, you can use the power rule of differentiation. The power rule states that if we have a function of the form:

${ f(x) = x^n }$

then the derivative of f with respect to x is:

${ f'(x) = nx^{n-1} }$

In this case, we have:

${ A = \frac{26.5}{x^{0.45}} }$

which can be rewritten as:

${ A = 26.5x^{-0.45} }$

Applying the power rule, we get:

${ \frac{dA}{dx} = -0.45 \cdot 26.5x^{-0.45-1} }$

${ \frac{dA}{dx} = -11.775x^{-1.45} }$

A: The derivative of A with respect to x represents the rate of change of A with respect to x. In other words, it measures how fast A changes when x changes.

A: No, the derivative of A with respect to x is not a function that can be used to find the value of A at a given x. The derivative is a measure of the rate of change of A with respect to x, not the value of A itself.

A: To round the answer to two decimal places, you can use the following steps:

Compute the value of the derivative at the given x.
Look at the third decimal place.
If the third decimal place is 5 or greater, round up. If it is less than 5, round down.

For example, if the value of the derivative at x = 0.23 is -50.2345, you would round it to -50.23.

A: The final answer is: -50.23

In this article, we answered some frequently asked questions related to the derivative of A with respect to x. We hope that this article has been helpful in clarifying any doubts you may have had about the derivative of A with respect to x.

The final answer is: -50.23