see explanation
Step-by-step explanation:
Using the chain rule and the standard derivatives
Given
y = f(g(x)) , then
= f'(g(x)) × g'(x) ← chain rule
(tanx) = sec²x ,
(cotx) = - csc²x
(c)
y = tan
= tan![x^{\frac{1}{2} }](/tpl/images/1062/4697/9a211.png)
= sec²
×
(
)
= sec²
× ![\frac{1}{2}](/tpl/images/1062/4697/9cdae.png)
![x^{-\frac{1}{2} }](/tpl/images/1062/4697/98190.png)
= sec²
× ![\frac{1}{2x^{\frac{1}{2} } }](/tpl/images/1062/4697/5bfd2.png)
= ![\frac{sec^2\sqrt{x} }{2\sqrt{x} }](/tpl/images/1062/4697/b18f7.png)
(d)
y = cot(1 + x)
= - csc²(1 + x) ×
(1 + x)
= - csc²(1 + x) × 1
= - csc²(1 + x)