4*x e ---------- 2*tan(4*x)
exp(4*x)/((2*tan(4*x)))
Apply the quotient rule, which is:
and .
To find :
Let .
The derivative of is itself.
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 :
The derivative of a constant times a function is the constant times the derivative of the function.
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:
So, the result is:
Now plug in to the quotient rule:
Now simplify:
The answer is:
/ 2 \ 4*x 1 4*x \-8 - 8*tan (4*x)/*e 4*----------*e + ----------------------- 2*tan(4*x) 2 4*tan (4*x)
/ / 2 \ / 2 \\ | 2*\1 + tan (4*x)/ / 2 \ | 1 + tan (4*x)|| 4*x 8*|1 - ----------------- + 2*\1 + tan (4*x)/*|-1 + -------------||*e | tan(4*x) | 2 || \ \ tan (4*x) // ----------------------------------------------------------------------- tan(4*x)
/ / 2 \\ | / 2 \ | 1 + tan (4*x)|| | 3 2 6*\1 + tan (4*x)/*|-1 + -------------|| | / 2 \ / 2 \ / 2 \ | 2 || | 1 2 6*\1 + tan (4*x)/ 3*\1 + tan (4*x)/ 10*\1 + tan (4*x)/ \ tan (4*x) /| 4*x 32*|-4 + -------- - 4*tan (4*x) - ------------------ - ----------------- + ------------------- + --------------------------------------|*e | tan(4*x) 4 2 2 tan(4*x) | \ tan (4*x) tan (4*x) tan (4*x) /