Final answer:
To find the implicit differentiation of the given expression, differentiate each term separately using the chain rule, then combine the derivatives.
Step-by-step explanation:
To find the implicit differentiation of the given expression, let's first differentiate each term separately. Starting with the first term, we have:
d/dx(sin^⁻¹(cos(3x))) = 1/(1 - cos^2(3x))(d/dx(cos(3x)))
Using the chain rule, d/dx(cos(3x)) = -3sin(3x). Applying the same process to the second term, we have:
d/dx(tan^⁻¹(12x)) = 1/(1 + (12x)^2)(d/dx(12x)) = 1/(1 + 144x^2)(12)
Combining these derivatives, we have:
d/dx(21sin^⁻¹(cos(3x))-tan^⁻¹(12x)) = 1/(1 - cos^2(3x))(-3sin(3x)) - 1/(1 + 144x^2)(12)