What is the derivative of ln({f}(x)(x)) ?

The derivative of ln({f}(x)(x)) is (x derivative of {f}(x))/({f)(x)}+ln({f}(x))

What is the first derivative of ln({f}(x)(x)) ?

The first derivative of ln({f}(x)(x)) is (x derivative of {f}(x))/({f)(x)}+ln({f}(x))

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\frac{d}{d x} (ln (f (x)) (x))

\frac{x \frac{d}{d x} ( f ( x ) )}{f ( x )} + ln (f (x))

Solution steps

\frac{d}{d x} (ln (f (x)) (x))

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = ln (f (x)), g = x

= \frac{d}{d x} (ln (f (x))) x + \frac{d x}{d x} ln (f (x))

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

\frac{d x}{d x} = 1

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

Simplify

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

= \frac{x \frac{d}{d x} ( f ( x ) )}{f ( x )} + ln (f (x))

y^{^{'}} = \frac{sec ( y )}{( x - 1 ) ^{2}}

\frac{d y}{d x} - \frac{2 y}{x} + x^{2} y^{2} = 0

t an g e n t f (x) = 19 x - x^{2}, at x = 9

x \to 2 - lim (- x + 10)

x \to 0 lim (\frac{x ^{4} - x ^{3}}{3})

What is the derivative of ln({f}(x)(x)) ?
The derivative of ln({f}(x)(x)) is (x derivative of {f}(x))/({f)(x)}+ln({f}(x))
What is the first derivative of ln({f}(x)(x)) ?
The first derivative of ln({f}(x)(x)) is (x derivative of {f}(x))/({f)(x)}+ln({f}(x))