HI, I really need some with this problem, please!

Question

Find the first and second derivatives of the following function y=x^3sin(x).

If you can, could you please explain how to solve it step by step. Thank you!

Answer 1

You need to use the Product Rule of Differentiation. Given the product of two functions f(x) and g(x) in the form

y = f(x) * g(x), then the derivative of y, which is written as y', can be found using the Product Rule, which is

f'(x) * g(x) + f(x) * g'(x), where f'(x) and g'(x) are the derivatives of f(x) and g(x), respectively.

Let f(x) = x³ and g(x) = sin(x)

You can find the derivative of f(x) using the Power Rule of Differentiation.

f'(x) = 3x²

The derivative of g(x) = cos(x)

Answer 2

Hello Almaz,

So this problem involves the product rule. Product rule is derivative of the first function times the second plus the derivative of the second function times the first. The first function in this case is x^3 so use the power rule to get the derivative of the first then multiply it by the second function itself. Now, you have the first part. Then, you need the derivative of the second function right? What is the derivative of sinx? Multiply this by the first function itself. Now you have the second part. Add these parts together for the first derivative. I hope this helps. If you need more help with derivatives, do not hesitate to contact me.

The second derivative will require you to take the individual derivatives of the first and second parts via product rule and add them together again. Just repeat the process.