Question

In finding derivatives with trigonometric substitutions how to identify when to substitute x with Cos Theta , or Tan Theta? . I tried to form following rule, please help me improve of correct " Always use x = Cos Theta unless you see a term with Positive X^2 or a Negative X2 in denominator below a numerator containing 2x or 3x " . Please help me improve above rule. ​

Answer 1

Answer:

Actually it is not necessary to consider cos© whenever you see an x^2.

For example y=(3x^2+1)^(1/3) can be differentiated using the basic formula

d/dx of (ax+b)^n= {n(ax+b)^(n-1) } . d/dx of (ax+b)

Similarly, y=ln {x-√(x^2-1)} can be differentiated by using usual formulas.

But in case of inverse trigonometric function you need to substitute x with such a ratio which will give you an advantage(you must know necessary trigonometric formulas). Such as

d/dx ( cos inverse (2x√(1-x^2)))

Here you can substitute x with cos© or sin©. For both cases the answer will be same but sign will be different .

If the function is tan inverse something then sometimes you can solve it without substitution, only by using the formula :

tan inverse {(a+b)/(1-ab)}= tan inverse (a) + tan inverse (b)