Final answer:
To find the derivative dy/dx of the function y=sec³(tan² 3x), you apply the chain rule and implicit differentiation, differentiate sec³(u) with respect to u, then multiply by the derivative of u=tan²(3x) with respect to x, and finally evaluate at u=tan²(3x).
Step-by-step explanation:
The student has asked to find the derivative of the function y=sec³(tan² 3x). This requires the use of chain rule and implicit differentiation techniques.
First, let's find dy/dx by differentiating both sides of the equation with respect to x. Using the chain rule, we differentiate sec³(u) with respect to u and then multiply by the derivative of u=tan²(3x) with respect to x. Finally, we differentiate tan²(3x) with respect to x.
The differentiation steps are as follows:
- Let u = tan²(3x). Then, y = sec³(u).
- Using the power rule and the chain rule, dy/du = 3sec²(u)sec(u)tan(u) = 3sec³(u)tan(u).
- The derivative of u with respect to x is du/dx = 2tan(3x)sec²(3x)⋅(3), because of the chain rule applied to tan²(3x).
- Therefore, dy/dx = dy/du ⋅ du/dx = 3sec³(u)tan(u) ⋅ 2tan(3x)sec²(3x)⋅(3).
Substitute back u = tan²(3x) into the final expression to obtain dy/dx in terms of x.
Keep in mind that finding this derivative involves understanding and applying advanced calculus operations like chain rule and implicit differentiation.