cos(x) 1 sin(x)*E + -------- tan(2*x)
Differentiate term by term:
Apply the product rule:
; to find :
The derivative of sine is cosine:
; to find :
Let .
The derivative of is itself.
Then, apply the chain rule. Multiply by :
The derivative of cosine is negative sine:
The result of the chain rule is:
The result is:
Let .
Apply the power rule: goes to
Then, apply the chain rule. Multiply by :
Rewrite the function to be differentiated:
Apply the quotient rule, which is:
and .
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:
To find :
Let .
The derivative of cosine is negative sine:
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:
Now plug in to the quotient rule:
The result of the chain rule is:
The result is:
Now simplify:
The answer is:
2 -2 - 2*tan (2*x) cos(x) 2 cos(x) ---------------- + cos(x)*e - sin (x)*e 2 tan (2*x)
2 / 2 \ / 2 \ 3 cos(x) cos(x) 8*\1 + tan (2*x)/ 8*\1 + tan (2*x)/ cos(x) sin (x)*e - e *sin(x) - ----------------- + ------------------ - 3*cos(x)*e *sin(x) tan(2*x) 3 tan (2*x)
3 2 / 2 \ / 2 \ 2 4 cos(x) cos(x) 48*\1 + tan (2*x)/ 2 cos(x) 2 cos(x) 80*\1 + tan (2*x)/ 2 cos(x) -32 - 32*tan (2*x) - sin (x)*e - cos(x)*e - ------------------- - 3*cos (x)*e + 4*sin (x)*e + ------------------- + 6*sin (x)*cos(x)*e 4 2 tan (2*x) tan (2*x)