_____________ / / 3 \ \/ cos\x + 4/
sqrt(cos(x^3 + 4))
Let .
Apply the power rule: goes to
Then, apply the chain rule. Multiply by :
Let .
The derivative of cosine is negative sine:
Then, apply the chain rule. Multiply by :
Differentiate term by term:
Apply the power rule: goes to
The derivative of the constant is zero.
The result is:
The result of the chain rule is:
The result of the chain rule is:
Now simplify:
The answer is:
2 / 3 \ -3*x *sin\x + 4/ ------------------ _____________ / / 3 \ 2*\/ cos\x + 4/
/ _____________ \ | / 3\ 3 / / 3\ 3 2/ 3\| | sin\4 + x / 3*x *\/ cos\4 + x / 3*x *sin \4 + x /| -3*x*|---------------- + --------------------- + -----------------| | _____________ 2 3/2/ 3\| | / / 3\ 4*cos \4 + x /| \\/ cos\4 + x / /
/ / 3\ _____________ 3 2/ 3\ 6 / 3\ 6 3/ 3\\ | sin\4 + x / 3 / / 3\ 9*x *sin \4 + x / 9*x *sin\4 + x / 27*x *sin \4 + x /| -3*|---------------- + 9*x *\/ cos\4 + x / + ----------------- + ------------------ + ------------------| | _____________ 3/2/ 3\ _____________ 5/2/ 3\ | | / / 3\ 2*cos \4 + x / / / 3\ 8*cos \4 + x / | \\/ cos\4 + x / 4*\/ cos\4 + x / /