Question
find the derivative of the function. y=[x+(x+sin^2(x)^5]^4
Answers
GPT 3.5
To find the derivative of the function y = [x + (x + sin^2(x)^5)^4], we will need to use the chain rule.
Let's break it down step by step:
1. First, let's find the derivative of the inner function (x + sin^2(x)^5)^4.
We can rewrite this as u^4, where u = (x + sin^2(x)^5).
Using the chain rule, the derivative of u^4 is 4u^3 times the derivative of u.
du/dx = 1 + 5sin^2(x)^4 * (2sin(x) * cos(x))
= 1 + 10sin^2(x)^4 * sin(x) * cos(x)
2. Now, we can find the derivative of y using the chain rule.
dy/dx = dy/du * du/dx
y = u^4, so dy/du = 4u^3
Substituting du/dx = 1 + 10sin^2(x)^4 * sin(x) * cos(x), we have:
dy/dx = 4u^3 * (1 + 10sin^2(x)^4 * sin(x) * cos(x))
= 4(x + sin^2(x)^5)^3 * (1 + 10sin^2(x)^4 * sin(x) * cos(x))
Therefore, the derivative of the function y = [x + (x + sin^2(x)^5)^4] is dy/dx = 4(x + sin^2(x)^5)^3 * (1 + 10sin^2(x)^4 * sin(x) * cos(x)).
Let's break it down step by step:
1. First, let's find the derivative of the inner function (x + sin^2(x)^5)^4.
We can rewrite this as u^4, where u = (x + sin^2(x)^5).
Using the chain rule, the derivative of u^4 is 4u^3 times the derivative of u.
du/dx = 1 + 5sin^2(x)^4 * (2sin(x) * cos(x))
= 1 + 10sin^2(x)^4 * sin(x) * cos(x)
2. Now, we can find the derivative of y using the chain rule.
dy/dx = dy/du * du/dx
y = u^4, so dy/du = 4u^3
Substituting du/dx = 1 + 10sin^2(x)^4 * sin(x) * cos(x), we have:
dy/dx = 4u^3 * (1 + 10sin^2(x)^4 * sin(x) * cos(x))
= 4(x + sin^2(x)^5)^3 * (1 + 10sin^2(x)^4 * sin(x) * cos(x))
Therefore, the derivative of the function y = [x + (x + sin^2(x)^5)^4] is dy/dx = 4(x + sin^2(x)^5)^3 * (1 + 10sin^2(x)^4 * sin(x) * cos(x)).