Final answer:
To evaluate (dy)/(dx) for y=cot⁻¹ (x) using implicit differentiation, we start by expressing y in terms of x and then take the derivative of both sides with respect to x. Finally, we solve for (dy)/(dx) to obtain the derivative expression in terms of x.
Step-by-step explanation:
To evaluate (dy)/(dx) for y=cot⁻¹ (x) using implicit differentiation, we start by expressing y in terms of x.
The inverse cotangent function is defined as y=cot⁻¹ (x), which means x=cot(y).
Taking the derivative of both sides with respect to x, we get dx=(-sin(y))/(1+cos²(y)) dy. Now we solve for (dy)/(dx) by rearranging the equation, giving us (dy)/(dx) = (-1)/(sin(y) (1+cos²(y))). Since x=cot(y), we can substitute x back into the equation to represent (dy)/(dx) entirely in terms of x, which is (dy)/(dx) = -1/(sin(cot⁻¹(x))(1+cos²(cot⁻¹(x)))).