Find the derivative of y = (x^2 + 1)(x^3 + 1) in two ways: by using the Product Rule and by performing the multiplicatino first. Do your answers agree?
I don't know if your answers agree. Can't you at least do the multiplication followed by differentiation?
For the product rule test, let u = x^2 +1 and v = x^3 +1.
Then d(uv)/dx = u dv/dx + v du/dx
= (x^2+1)*3x^2 + (x^3+1)*2x
= 3x^4 + 3x^2 + 2x^4 + 2x
= 5x^4 + 3x^2 + 2x
Now do it the other way and see of they agree
To find the derivative of y = (x^2 + 1)(x^3 + 1), we can use two different approaches: the Product Rule and multiplying first. Let's go step by step for both methods and see if the answers agree.
Approach 1: Using the Product Rule
The Product Rule states that if we have two functions, u(x) and v(x), their product u(x) * v(x) can be differentiated as follows:
d/dx(u(x) * v(x)) = u(x) * [d/dx(v(x))] + v(x) * [d/dx(u(x))]
In our case, u(x) = (x^2 + 1) and v(x) = (x^3 + 1). Let's find the derivatives first.
Taking the derivative of u(x):
d/dx(x^2 + 1) = 2x
Taking the derivative of v(x):
d/dx(x^3 + 1) = 3x^2
Now we can apply the Product Rule:
d/dx[(x^2 + 1)(x^3 + 1)] = (x^2 + 1) * (3x^2) + (x^3 + 1) * (2x)
= 3x^4 + 3x^2 + 2x^4 + 2x
= 5x^4 + 3x^2 + 2x
Approach 2: Multiplying First
We can multiply the two terms in the expression y = (x^2 + 1)(x^3 + 1) first and then find the derivative of the resulting polynomial.
Expanding the product:
y = x^5 + x^3 + x^2 + 1
Taking the derivative of y:
d/dx(x^5 + x^3 + x^2 + 1) = 5x^4 + 3x^2 + 2x
Upon comparing the results from both approaches, we can observe that the derivatives obtained using the Product Rule and multiplying first are indeed the same:
Result from the Product Rule: 5x^4 + 3x^2 + 2x
Result from multiplying first: 5x^4 + 3x^2 + 2x
Therefore, both methods yield the same derivative, confirming the correctness of the answers.