differentiate
f(x)=(1+sinˆ2x)ˆ6
if f=u^n,
f' = n u^(n-1) u'
so, whatcha got?
once again the formula doesnt help me if i don't understand the identities.
To differentiate the function f(x) = (1 + sin²x)⁶, we can apply the chain rule, as well as the derivative of the sin²x function.
Step 1: Apply the Chain Rule
The chain rule states that if we have a composite function, such as f(g(x)), the derivative can be found by taking the derivative of the outer function, multiplied by the derivative of the inner function.
Let's denote the inner function as g(x) = 1 + sin²x, and the outer function as h(u) = u⁶. So, f(x) = h(g(x)).
Step 2: Find the Derivative of g(x)
To find the derivative of g(x), we need to use the chain rule again, as g(x) is a composite function.
g(x) = 1 + sin²x
Let's denote the inner function as u(x) = sin²x, and the outer function as v(u) = 1 + u.
The derivative of u(x) = sin²x can be found using the chain rule as follows:
du/dx = 2sin(x)cos(x)
Now, we can find the derivative of g(x) using the chain rule:
dg/dx = dv(du/dx)
= dv(2sin(x)cos(x))
= 2cos(x)dv/dx
Since dv/dx = d(1 + u)/du * du/dx = 1 * 2sin(x)cos(x) = 2sin(x)cos(x), we have:
dg/dx = 2cos(x) * 2sin(x)cos(x)
= 4sin(x)cos²(x)
Step 3: Find the Derivative of h(u)
To find the derivative of h(u) = u⁶, we need to apply the power rule.
The power rule states that if we have a function f(u) = u^n, then its derivative is given by:
df/du = n * u^(n-1)
In this case, h(u) = u⁶, so we have:
dh/du = 6 * u^(6-1)
= 6u⁵
Step 4: Find the Derivative of f(x)
Now, we can find the derivative of the original function f(x) = h(g(x)) using the chain rule.
df/dx = dh/du * dg/dx
= 6u⁵ * 4sin(x)cos²(x)
Since u = g(x) = 1 + sin²x, we have:
df/dx = 6(1 + sin²x)⁵ * 4sin(x)cos²(x)
Therefore, the derivative of f(x) = (1 + sin²x)⁶ is 6(1 + sin²x)⁵ * 4sin(x)cos²(x).