Find the derivatives

Question

Find dy/dx:

1.) y = (tanx) / (1+xtanx)

2.) y = cot(x^2)+sec^2x

3.) x^3 -xy +y^2 =4

4.) y = xarctanx

Answer 1

There are some things you need to know to do these problems that, hopefully you've learned.

1) Basic derivatives:

a) the derivative of tan(x) is sec²(x)

b) the derivative of cot(x) is – csc²(x)

c) the derivative of sec(x) is sec(x)tan(x)

d) the derivative of arctan(x) is 1/(1+x²)

2) The product rule and the quotient rule

a) product rule: (fg)' = f'g + g'f

b) quotient rule: (f/g)' = (f'g - g'f)/g²

3) The chain rule: (f(g(x))' = f'(g(x))•g'(x)

4) Implicit differentiation (same as the chain rule basically)

The first problem uses both 1a and 2a and 2b. Since it's a division, the basic rule you use is the quotient rule. But to take the derivative of the denominator, you must use the product rule as well.

I'm somewhat concerned if you are unable to do these problems. Perhaps a tutoring session would be worthwhile. I'd be happy to meet you for an initial session free of charge.