4 4 sec (x) - tan (x)
sec(x)^4 - tan(x)^4
Differentiate term by term:
Let .
Apply the power rule: goes to
Then, apply the chain rule. Multiply by :
Rewrite the function to be differentiated:
Let .
Apply the power rule: goes to
Then, apply the chain rule. Multiply by :
The derivative of cosine is negative sine:
The result of the chain rule is:
The result of the chain rule is:
The derivative of a constant times a function is the constant times the derivative of the function.
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 :
The derivative of sine is cosine:
To find :
The derivative of cosine is negative sine:
Now plug in to the quotient rule:
The result of the chain rule is:
So, the result is:
The result is:
Now simplify:
The answer is:
3 / 2 \ 4 - tan (x)*\4 + 4*tan (x)/ + 4*sec (x)*tan(x)
/ 2 \ | 4 / 2 \ / 2 \ 2 4 / 2 \ 4 2 | 4*\sec (x)*\1 + tan (x)/ - 3*\1 + tan (x)/ *tan (x) - 2*tan (x)*\1 + tan (x)/ + 4*sec (x)*tan (x)/
/ 3 2 \ | / 2 \ / 2 \ 2 4 / 2 \ 4 / 2 \ 4 2 | 8*\- 3*\1 + tan (x)/ - 10*\1 + tan (x)/ *tan (x) - 2*tan (x)*\1 + tan (x)/ + 7*sec (x)*\1 + tan (x)/ + 8*sec (x)*tan (x)/*tan(x)