cos(sin(3*x)) ---------------- / _________\ tan\\/ 2*x + 1 /
cos(sin(3*x))/tan(sqrt(2*x + 1))
Apply the quotient rule, which is:
and .
To find :
Let .
The derivative of cosine is negative sine:
Then, apply the chain rule. Multiply by :
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:
The result of the chain rule is:
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 :
Let .
Apply the power rule: goes to
Then, apply the chain rule. Multiply by :
Differentiate term by term:
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 derivative of the constant is zero.
The result is:
The result of the chain rule 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 :
Let .
Apply the power rule: goes to
Then, apply the chain rule. Multiply by :
Differentiate term by term:
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 derivative of the constant is zero.
The result is:
The result of the chain rule is:
The result of the chain rule is:
Now plug in to the quotient rule:
Now plug in to the quotient rule:
Now simplify:
The answer is:
/ 2/ _________\\ 3*cos(3*x)*sin(sin(3*x)) \1 + tan \\/ 2*x + 1 //*cos(sin(3*x)) - ------------------------ - ------------------------------------- / _________\ _________ 2/ _________\ tan\\/ 2*x + 1 / \/ 2*x + 1 *tan \\/ 2*x + 1 /
/ / 2/ _________\\ \ / 2/ _________\\ 2 / 2/ _________\\ | 2 1 2*\1 + tan \\/ 1 + 2*x // | 6*\1 + tan \\/ 1 + 2*x //*cos(3*x)*sin(sin(3*x)) - 9*cos (3*x)*cos(sin(3*x)) + 9*sin(3*x)*sin(sin(3*x)) + \1 + tan \\/ 1 + 2*x //*|- ------- + ----------------------------- + ---------------------------|*cos(sin(3*x)) + ------------------------------------------------ | 1 + 2*x 3/2 / _________\ 2/ _________\| _________ / _________\ \ (1 + 2*x) *tan\\/ 1 + 2*x / (1 + 2*x)*tan \\/ 1 + 2*x // \/ 1 + 2*x *tan\\/ 1 + 2*x / --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- / _________\ tan\\/ 1 + 2*x /
/ / 2/ _________\\ \ / 2/ _________\\ | 2 1 2*\1 + tan \\/ 1 + 2*x // | / 2 \ 9*\1 + tan \\/ 1 + 2*x //*|- ------- + ----------------------------- + ---------------------------|*cos(3*x)*sin(sin(3*x)) | / 2/ _________\\ / 2/ _________\\ / 2/ _________\\ | / 2 \ / 2/ _________\\ / 2 \ | 1 + 2*x 3/2 / _________\ 2/ _________\| / 2/ _________\\ | 4 6 3 10*\1 + tan \\/ 1 + 2*x // 6*\1 + tan \\/ 1 + 2*x // 6*\1 + tan \\/ 1 + 2*x // | 27*\cos (3*x)*sin(sin(3*x)) + 3*cos(sin(3*x))*sin(3*x) + sin(sin(3*x))/*cos(3*x) 27*\1 + tan \\/ 1 + 2*x //*\sin(3*x)*sin(sin(3*x)) - cos (3*x)*cos(sin(3*x))/ \ (1 + 2*x) *tan\\/ 1 + 2*x / (1 + 2*x)*tan \\/ 1 + 2*x // - \1 + tan \\/ 1 + 2*x //*|------------ - --------------------------- + ------------------------------ - ------------------------------ + ------------------------------ + ----------------------------|*cos(sin(3*x)) + -------------------------------------------------------------------------------- - ----------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------- | 3/2 2 / _________\ 5/2 2/ _________\ 3/2 2/ _________\ 3/2 4/ _________\ 2 3/ _________\| / _________\ _________ 2/ _________\ / _________\ \(1 + 2*x) (1 + 2*x) *tan\\/ 1 + 2*x / (1 + 2*x) *tan \\/ 1 + 2*x / (1 + 2*x) *tan \\/ 1 + 2*x / (1 + 2*x) *tan \\/ 1 + 2*x / (1 + 2*x) *tan \\/ 1 + 2*x // tan\\/ 1 + 2*x / \/ 1 + 2*x *tan \\/ 1 + 2*x / tan\\/ 1 + 2*x /