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The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If $f(x)=ln\:a$ (where $a$ is a function of $x$), then $\displaystyle f'(x)=\frac{a'}{a}$
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$\frac{1}{\sec\left(\frac{1}{x}\right)}\frac{d}{dx}\left(\sec\left(\frac{1}{x}\right)\right)$
Learn how to solve differential calculus problems step by step online. Find the derivative of ln(sec(1/x)). The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If f(x)=ln\:a (where a is a function of x), then \displaystyle f'(x)=\frac{a'}{a}. Taking the derivative of secant function: \frac{d}{dx}\left(\sec(x)\right)=\sec(x)\cdot\tan(x)\cdot D_x(x). Apply the quotient rule for differentiation, which states that if f(x) and g(x) are functions and h(x) is the function defined by {\displaystyle h(x) = \frac{f(x)}{g(x)}}, where {g(x) \neq 0}, then {\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}. Multiplying fractions \frac{1}{\sec\left(\frac{1}{x}\right)} \times \frac{\frac{d}{dx}\left(1\right)x-\frac{d}{dx}\left(x\right)}{x^2}.