Question

How do you differentiate `y= (sin^2 x tan^4 x) / (x^2 + 1)^2`?

Answer 1

The answer is long. Please see below.

Explanation:

Since this is a quotient we know we have to use the quotient rule. But to do that, we need to know the derivatives of the top and the bottom.

Additionally, the top itself is a product of two separate functions. in order to calculate the derivative of the top we will need the derivatives of each function. Therefore, let's first find the derivatives of each piece:

`d/dx(sin^2x)=2sinxcosx`

`d/dx(tan^4x)=4tan^3xsec^2x`

`d/dx(sin^2xtan^4x)=sin^2x(4tan^3xsec^2x)+tan^4x(2sinxcosx)=4sin^2xsin^3x/(cos^3xcos^2x)+2(sin^4x/cos^4x)sinxcos=4sin^5x/cos^5x+2sin^5x/cos^3x=4tan^5x+2tan^3xsin^2x`

`d/dx(x^2+1)^2=2(x^2+1)(2x)=4x(x^2+1)`

Now using the quotient rule we have:

`dy/dx=(((x^2+1)^2(4tan^5x+2tan^3xsin^2x)-sin^2xtan^4x(4x(x^2+1)))/(x^2+1)^4)`

Using some algebra and factoring we get:

`dy/dx=((2(x^2+1)tan^3x(2tan^2x+sin^2x)-4xsin^2xtan^4x)/(x^2+1)^3)`

There are many different forms for this answer depending on what trig. formulas you use and how you simplify.