find d/dx [cot(sec^3(cos(3x^2-5x+3)))]
To find the derivative of the function cot(sec^3(cos(3x^2-5x+3)), we need to apply the chain rule and the derivative rules for trigonometric functions.
Let's break it down step by step:
Step 1: Identify the function and its composition
The function we need to differentiate is cot(sec^3(cos(3x^2-5x+3))). We can see that it is composed of three functions:
1. Outer function: cot(x)
2. Middle function: sec^3(x)
3. Inner function: cos(3x^2-5x+3)
Step 2: Apply the chain rule
The chain rule states that if we have a composition of functions f(g(x)), then the derivative is given by d/dx[f(g(x))] = f'(g(x)) * g'(x).
Applying the chain rule to our problem:
d/dx [cot(sec^3(cos(3x^2-5x+3)))]
= (-csc^2(sec^3(cos(3x^2-5x+3)))) * (d/dx [sec^3(cos(3x^2-5x+3))])
Step 3: Differentiate the outer function
The derivative of cot(x) is -csc^2(x). So, we substitute cot(sec^3(cos(3x^2-5x+3))) with (-csc^2(sec^3(cos(3x^2-5x+3)))).
Step 4: Differentiate the inner function
Now we need to find the derivative of the inner function. Let's break it down again:
Inner function: cos(3x^2-5x+3)
Derivative of inner function: -sin(3x^2-5x+3) * (d/dx [3x^2-5x+3])
Step 5: Differentiate the remaining terms
Let's differentiate the remaining terms from the derivative of the inner function:
d/dx [3x^2-5x+3]
= (d/dx [3x^2]) + (d/dx [-5x]) + (d/dx [3])
= 6x - 5
Step 6: Substitute back into the derivative expression
Now we can substitute the derivative of the inner function into the expression we found in Step 2:
d/dx [cot(sec^3(cos(3x^2-5x+3)))]
= (-csc^2(sec^3(cos(3x^2-5x+3)))) * (-sin(3x^2-5x+3) * (6x - 5))
Simplifying further if needed is possible, but this is the derivative expression using the chain rule and derivative rules for trigonometric functions.