derivative of ln(ln(sec(2x))) — step-by-step solution

Let's differentiate:

\frac{d}{d x} (\ln (\ln (\sec (2 x))))

Apply the differentiation rule:

\frac{d}{d x} (\ln (f)) = \frac{d}{d x} (f) \cdot \frac{1}{f}

with f =

\ln (\sec (2 x))

ⓘ The derivative of ln(x) is 1/x:

\frac{d}{d x} (\ln (\ln (\sec (2 x))))

\frac{d}{d x} (\ln (\sec (2 x))) \cdot \frac{1}{\ln (\sec (2 x))}

\frac{d}{d x} (\ln (\sec (2 x))) \cdot \frac{1}{\ln (\sec (2 x))}

Apply the differentiation rule:

\frac{d}{d x} (\ln (f)) = \frac{d}{d x} (f) \cdot \frac{1}{f}

with f =

\sec (2 x)

ⓘ Multiplying a fraction by a term multiplies its numerator:

\frac{d}{d x} (\ln (\sec (2 x))) \cdot \frac{1}{\ln (\sec (2 x))}

\frac{\frac{d}{d x} (\ln (\sec (2 x))) \cdot 1}{\ln (\sec (2 x))}

\frac{\frac{d}{d x} (\ln (\sec (2 x)))}{\ln (\sec (2 x))}

■The derivative of ln(x) is 1/x:

\frac{d}{d x} (\ln (\sec (2 x)))

\frac{d}{d x} (\sec (2 x)) \cdot \frac{1}{\sec (2 x)}

\frac{\frac{d}{d x} (\sec (2 x)) \cdot \frac{1}{\sec (2 x)}}{\ln (\sec (2 x))}

Apply the differentiation rule:

\frac{d}{d x} (\sec (f)) = \frac{d}{d x} (f) \cdot \sec (f) \cdot \tan (f)

with f =

2 x

ⓘ Multiplying a fraction by a term multiplies its numerator:

\frac{d}{d x} (\sec (2 x)) \cdot \frac{1}{\sec (2 x)}

\frac{\frac{d}{d x} (\sec (2 x)) \cdot 1}{\sec (2 x)}

\frac{\frac{d}{d x} (\sec (2 x))}{\sec (2 x)}

■The derivative of sec(x) is sec(x)*tan(x):

\frac{d}{d x} (\sec (2 x))

\frac{d}{d x} (2 x) \cdot \sec (2 x) \cdot \tan (2 x)

\frac{\frac{\frac{d}{d x} (2 x) \cdot \sec (2 x) \cdot \tan (2 x)}{\sec (2 x)}}{\ln (\sec (2 x))}

ⓘ The original denominator must remain nonzero. The fraction is simplified by dividing the numerator and the denominator by their common factor

\sec (2 x)

\frac{\frac{d}{d x} (2 x) \cdot \sec (2 x) \cdot \tan (2 x)}{\sec (2 x)}

\frac{d}{d x} (2 x) \cdot \tan (2 x)

\frac{\frac{d}{d x} (2 x) \cdot \tan (2 x)}{\ln (\sec (2 x))}

ⓘ A constant factor stays in front of the derivative, (k*u)' = k*u':

\frac{d}{d x} (2 x)

2

▶Derivative:

\frac{2 \cdot \tan (2 x)}{\ln (\sec (2 x))}