differentiate with respect to x

Question

a) y=e^-x²    b)   y=e^tanx    c)  y=e^2x²

Answer 1

These derivatives can by worked out using the chain rule:   derivative of  g( f(x) ) =  f'(x)  g'( f(x) ) 

 

For a)   the derivative of e^x   is just e^x  itself.    So the derivative of  e^{-x ^2}     is e^{-x^2}  times the derivative of        - x^2.

  The derivative  of      - x^2   is   - 2 x.  

  So the final answer is    - 2 x e^{-x^2}

 

For b)  The derivative of  e^x  is again  e^x .   So the derivative of  e^tanx  is  e^tanx  times the derivative of tanx.  

   The derivative of tanx  is    sec²x.

  So the final answer is      sec²x e^tanx .

 

For c)  the procedure is similar to that for a).  Instead of a factor of a factor of -1 in the exponential, there is a factor of 2.

   So the final answer is  4 x  e^{2x^2}  .