1 --------------- / 3\ sin\3*x - 4*x /
1/sin(3*x - 4*x^3)
Let .
Apply the power rule: goes to
Then, apply the chain rule. Multiply by :
Let .
The derivative of sine is cosine:
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 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 is:
The result of the chain rule is:
The result of the chain rule is:
Now simplify:
The answer is:
/ 2\ / 3\ -\3 - 12*x /*cos\-3*x + 4*x / ------------------------------ 2/ 3\ sin \3*x - 4*x /
/ 2 \ | 2 / 2\ 2/ / 2\\ / / 2\\| | / 2\ 6*\-1 + 4*x / *cos \x*\-3 + 4*x // 8*x*cos\x*\-3 + 4*x //| 3*|- 3*\-1 + 4*x / - ---------------------------------- + ----------------------| | 2/ / 2\\ / / 2\\ | \ sin \x*\-3 + 4*x // sin\x*\-3 + 4*x // / ---------------------------------------------------------------------------------- / / 2\\ sin\x*\-3 + 4*x //
/ 3 3 \ | / / 2\\ / 2\ / / 2\\ / 2\ 3/ / 2\\ 2/ / 2\\ / 2\| | / 2\ 8*cos\x*\-3 + 4*x // 45*\-1 + 4*x / *cos\x*\-3 + 4*x // 54*\-1 + 4*x / *cos \x*\-3 + 4*x // 144*x*cos \x*\-3 + 4*x //*\-1 + 4*x /| 3*|- 72*x*\-1 + 4*x / + -------------------- + ---------------------------------- + ----------------------------------- - -------------------------------------| | / / 2\\ / / 2\\ 3/ / 2\\ 2/ / 2\\ | \ sin\x*\-3 + 4*x // sin\x*\-3 + 4*x // sin \x*\-3 + 4*x // sin \x*\-3 + 4*x // / ---------------------------------------------------------------------------------------------------------------------------------------------------------------- / / 2\\ sin\x*\-3 + 4*x //