For some equations like x^2+y^2=1, it is possible to solve for y and then calculate dy/dx. even in these cases, explain why implicit differentiation is usually a more efficient method for calculating the derivative.

The solution to some implicit equations might be multiple functions, like in the example in your question:

and so you have to treat them separately when finding the derivative (this can be much more messy than in the example above)

Instead, it is more efficient to do the differentiation implicitly - once. Lmk if you have questions.

a box 3 3/4 feet long or 4+ feet.

step-by-step explanation:

x= -3 and

step-by-step explanation:

to find the roots of a cubic, use division or factoring to find the factors then set equal to 0. as this is a difference of cubes, it factors into

where a is the cube root of .

here   this is 27 and the cube root of 27 is 3. so we write using the form:

.

set each factor to 0 and solve for x.

x-3=0

x=3

the quadratic expression does not factor and must be solved using the quadratic formula.

where a = 1, b=3, and c=9

this cubic has two complex roots.