What is the derivative of log_{x}(y) ?

The derivative of log_{x}(y) is (xln(x)\frac{dy)/(dx)-ln(y)y}{xln^2(x)y}

What is the first derivative of log_{x}(y) ?

The first derivative of log_{x}(y) is (xln(x)\frac{dy)/(dx)-ln(y)y}{xln^2(x)y}

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\frac{d}{d x} (lo g_{x} (y))

\frac{x ln ( x ) \frac{d y}{d x} - ln ( y ) y}{x ln ^{2} ( x ) y}

Solution steps

\frac{d}{d x} (lo g_{x} (y))

Apply log rule:

lo g_{a} (b) = \frac{ln ( b )}{ln ( a )}

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

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{d}{d x} ( ln ( y ) ) ln ( x ) - \frac{d}{d x} ( ln ( x ) ) ln ( y )}{( ln ( x ) ) ^{2}}

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

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

= \frac{\frac{\frac{d y}{d x}}{y} ln ( x ) - \frac{1}{x} ln ( y )}{( ln ( x ) ) ^{2}}

Simplify

\frac{\frac{\frac{d y}{d x}}{y} ln ( x ) - \frac{1}{x} ln ( y )}{ln ^{2} ( x )} : \frac{x ln ( x ) \frac{d y}{d x} - ln ( y ) y}{x ln ^{2} ( x ) y}

= \frac{x ln ( x ) \frac{d y}{d x} - ln ( y ) y}{x ln ^{2} ( x ) y}

\int 6 x^{2} - 4 x + 8 d x

\frac{d y}{d x} = (3 + y)^{2}

t \to \infty lim (800 - 800 t^{- 1})

g (x) = \frac{1}{x + 1}

\int_{0}^{\frac{π}{2}} 2 sin (2 x) d x

What is the derivative of log_{x}(y) ?
The derivative of log_{x}(y) is (xln(x)\frac{dy)/(dx)-ln(y)y}{xln^2(x)y}
What is the first derivative of log_{x}(y) ?
The first derivative of log_{x}(y) is (xln(x)\frac{dy)/(dx)-ln(y)y}{xln^2(x)y}