/ / 3 \\ sin\log\x + 1//
d / / / 3 \\\ --\sin\log\x + 1/// dx
Let .
The derivative of sine is cosine:
Then, apply the chain rule. Multiply by :
Let .
The derivative of is .
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 *cos\log\x + 1// --------------------- 3 x + 1
/ 3 / / 3\\ 3 / / 3\\\ | / / 3\\ 3*x *cos\log\1 + x // 3*x *sin\log\1 + x //| 3*x*|2*cos\log\1 + x // - --------------------- - ---------------------| | 3 3 | \ 1 + x 1 + x / ------------------------------------------------------------------------ 3 1 + x
/ 3 / / 3\\ 3 / / 3\\ 6 / / 3\\ 6 / / 3\\\ | / / 3\\ 18*x *cos\log\1 + x // 18*x *sin\log\1 + x // 9*x *cos\log\1 + x // 27*x *sin\log\1 + x //| 3*|2*cos\log\1 + x // - ---------------------- - ---------------------- + --------------------- + ----------------------| | 3 3 2 2 | | 1 + x 1 + x / 3\ / 3\ | \ \1 + x / \1 + x / / ------------------------------------------------------------------------------------------------------------------------- 3 1 + x