_________ \/ 3*x + 1 *sin(2*x)
d / _________ \ --\\/ 3*x + 1 *sin(2*x)/ dx
Apply the product rule:
; to find :
Let .
Apply the power rule: goes to
Then, apply the chain rule. Multiply by :
Differentiate term by term:
The derivative of a constant times a function is the constant times the derivative of the function.
Apply the power rule: goes to
So, the result is:
The derivative of the constant is zero.
The result is:
The result of the chain rule is:
; to find :
Let .
The derivative of sine is cosine:
Then, apply the chain rule. Multiply by :
The derivative of a constant times a function is the constant times the derivative of the function.
Apply the power rule: goes to
So, the result is:
The result of the chain rule is:
The result is:
Now simplify:
The answer is:
_________ 3*sin(2*x) 2*\/ 3*x + 1 *cos(2*x) + ------------- _________ 2*\/ 3*x + 1
_________ 6*cos(2*x) 9*sin(2*x) - 4*\/ 1 + 3*x *sin(2*x) + ----------- - -------------- _________ 3/2 \/ 1 + 3*x 4*(1 + 3*x)
18*sin(2*x) _________ 27*cos(2*x) 81*sin(2*x) - ----------- - 8*\/ 1 + 3*x *cos(2*x) - -------------- + -------------- _________ 3/2 5/2 \/ 1 + 3*x 2*(1 + 3*x) 8*(1 + 3*x)