What is the first derivative of sin^4(x^2+1)?

We'll use the chain rule to differentiate the given
function:

f'(x) = {[sin(x^2 +
1)]^4}'

We'll differentiate applying the power rule first,
then we'll differentiate the sine function and, in the end, we'll differentiate the
expression x^2 + 1.

f'(x) = 4[sin(x^2 +1)]^3*[cos(x^2
+1)]*(2x)

f'(x) = 8x[sin(x^2 +1)]^3*[cos(x^2
+1)]

We can re-write [sin(x^2 +1)]^2 = 1 -
[cos(x^2 +1)]^2

f'(x) = 8x[sin(x^2 +1)]*[cos(x^2 +1)]*{1 -
[cos(x^2 +1)]^2}

f'(x) = 4x*sin2(x^2 +1)*{1 -
[cos(x^2 +1)]^2}