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).