To determine

To find: The derivative of y=cos(sec4x).

Answer

The derivative of y=cos(sec4x) is −4sin(sec4x)sec(4x)tan(4x).

Explanation

Derivative Rules:

The Chain Rule:

If g is differentiable at x and f is differentiable at g(x), then the composite function F=f∘g defined by F(x)=f(g(x)) is differentiable at x and F′ is given by the product

F′(x)=f′(g(x))⋅g′(x) (1)

Calculation:

Obtain the derivative of y.

y′=ddx(cos(sec4x))

Apply chain rule and simplify as,

y′=(−sin(sec4x))ddx(sec4x)

Apply chain rule once again and simplify as,

y′=(−sin(sec4x))ddx(sec4x)=(−sin(sec4x))(sec4xtan4x)ddx(4x)=−sin(sec4x)(sec4xtan4x)(4)=−4sin(sec4x)sec(4x)tan(4x)

Therefore, the derivative of y=cos(sec4x) is −4sin(sec4x)sec(4x)tan(4x).