/ ___\ /1\ x*tan\2*\/ x / + 7*sin|-| \x/
x*tan(2*sqrt(x)) + 7*sin(1/x)
Differentiate term by term:
Apply the product rule:
; to find :
Apply the power rule: goes to
; to find :
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 is:
The derivative of a constant times a function is the constant times the derivative of the function.
Let .
The derivative of sine is cosine:
Then, apply the chain rule. Multiply by :
Apply the power rule: goes to
The result of the chain rule is:
So, the result is:
The result is:
Now simplify:
The answer is:
/1\ 7*cos|-| ___ / 2/ ___\\ \x/ / ___\ \/ x *\1 + tan \2*\/ x // - -------- + tan\2*\/ x / 2 x
/1\ /1\ 7*sin|-| 14*cos|-| / 2/ ___\\ \x/ / 2/ ___\\ / ___\ \x/ 3*\1 + tan \2*\/ x // - -------- + 2*\1 + tan \2*\/ x //*tan\2*\/ x / + --------- + --------------------- 4 3 ___ x x 2*\/ x
/1\ 2 /1\ /1\ 42*cos|-| / 2/ ___\\ 7*cos|-| 42*sin|-| / 2/ ___\\ / 2/ ___\\ / ___\ 2/ ___\ / 2/ ___\\ \x/ 2*\1 + tan \2*\/ x // \x/ \x/ 3*\1 + tan \2*\/ x // 3*\1 + tan \2*\/ x //*tan\2*\/ x / 4*tan \2*\/ x /*\1 + tan \2*\/ x // - --------- + ---------------------- + -------- + --------- - --------------------- + ---------------------------------- + ----------------------------------- 4 ___ 6 5 3/2 x ___ x \/ x x x 4*x \/ x