# Derivatives and order of operations/rules

In summary, the conversation discussed difficulties with derivatives and whether to use the chain rule or product rule first. It was suggested to use the power rule and to remember that the derivative of a sum is the sum of the derivatives. An example problem was also provided, demonstrating the use of both the chain and product rule.
Hi, I am having troubles with derivitives like should i use chain rule first before using product rule and such.

heres an example problem:

3(1-5x)^1/2 + 1/6(1-5x)^3/2

What should my following steps be?

Remember that the derivative of a sum is the sum of the derivatives, the derivative of c f(x) is c times the derivative of f(x) when c is a constant, the power rule and to use the chain rule when finding the derivative of (1-5x)^(1/2) etc.

Hope that helps!

Last edited:
aznclink, the following example is one where you have to use both chain and product rule:
differentiate: y = (2x+1)^5 * (x^3-x+1)^4.
In this example you must first use the product rule, y=f(x)g'(x) + g(x)f'(x), and then the chain rule to find g'(x) and f'(x).

hope this helps.

## 1. What are derivatives and why are they important in mathematics?

Derivatives are a mathematical tool used to find the rate of change of a function at a specific point. They are important in mathematics because they allow us to analyze and understand the behavior of functions, which has numerous applications in fields such as physics, engineering, and economics.

## 2. What is the order of operations and why is it important when working with derivatives?

The order of operations refers to the specific sequence in which mathematical operations should be performed. In the context of derivatives, it is important to follow the order of operations to ensure accurate and consistent results. This is especially crucial when dealing with complex functions involving multiple operations.

## 3. What are the basic rules for finding derivatives?

The basic rules for finding derivatives include the power rule, product rule, quotient rule, and chain rule. The power rule states that the derivative of a function raised to a power is equal to the power multiplied by the function raised to the power minus one. The product rule states that the derivative of a product of two functions is equal to the first function times the derivative of the second function, plus the second function times the derivative of the first function. The quotient rule states that the derivative of a quotient of two functions is equal to the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, all divided by the denominator squared. The chain rule is used to find the derivative of a composite function, and it states that the derivative is equal to the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.

## 4. How are derivatives used to find maximum and minimum values of a function?

Derivatives are used to find the maximum and minimum values of a function by setting the derivative equal to zero and solving for the critical points. These critical points represent points where the slope of the function is either zero or undefined, which can indicate a maximum or minimum value. The second derivative is then used to determine if the critical point is a maximum or minimum by evaluating the concavity of the function at that point.

## 5. Are there any real-life applications of derivatives?

Yes, derivatives have numerous real-life applications. For example, they are used in physics to calculate the velocity and acceleration of moving objects, in economics to determine the marginal cost and revenue of production, and in engineering to analyze the rate of change of various physical quantities. They are also used in fields such as medicine, biology, and finance to model and predict various phenomena.

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