# Taylor series problem (non-direct differentiation?)

• Loopas
In summary, the speaker attached a picture of a problem from their online homework. They know how to solve it through direct differentiation, but that would take too long for this problem. The problem suggests finding another way, so they are asking for the best way to solve it. The solution is to multiply the Taylor series of \sin(7x) by x^3. The speaker thanks the person for their help.
Loopas
I attached a picture of the problem from my online HW. I know how to solve the problem through direct differentiation, but that would too long to find the derivatives for this problem, and the problem actually suggests that I find another way. So my question is, what's the best way to solve this?

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• hw.jpg
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Loopas said:
I attached a picture of the problem from my online HW. I know how to solve the problem through direct differentiation, but that would too long to find the derivatives for this problem, and the problem actually suggests that I find another way. So my question is, what's the best way to solve this?

Multiply the Taylor series of $\sin(7x)$ by $x^3$.

Thank you... wouldve never guessed it was so simple.

## 1. What is a Taylor series?

A Taylor series is a mathematical representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point. It is used to approximate a complex function with a simpler polynomial function, making it easier to work with.

## 2. What is the purpose of a Taylor series?

The purpose of a Taylor series is to approximate a complex function with a simpler polynomial function, allowing for easier calculations and analysis. It can also be used to find the value of a function at a point where the original function is not defined.

## 3. How is a Taylor series calculated?

A Taylor series is calculated by taking derivatives of a function at a specific point and plugging them into a formula that involves the function's value at that point. The more derivatives that are included in the calculation, the more accurate the Taylor series approximation will be.

## 4. What is the difference between direct and non-direct differentiation in a Taylor series?

In direct differentiation, the derivatives of a function are explicitly calculated and used in the Taylor series formula. In non-direct differentiation, the derivatives are not explicitly calculated, but are found by using other methods such as integration or power series expansion.

## 5. How can a Taylor series be used for non-direct differentiation?

A Taylor series can be used for non-direct differentiation by finding the derivatives of a function using other methods such as integration or power series expansion, and then plugging those derivatives into the Taylor series formula. This allows for more complex functions to be approximated with a Taylor series.

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