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Popular >

derivative of ln(y/x)

Solution

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

Solution

\frac{x \frac{d y}{d x} - y}{x y}

Solution steps

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

Apply the chain rule:

\frac{1}{\frac{y}{x}} \frac{d}{d x} (\frac{y}{x})

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

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

= \frac{1}{\frac{y}{x}} \cdot \frac{x \frac{d y}{d x} - y}{x ^{2}}

Simplify

\frac{1}{\frac{y}{x}} \cdot \frac{x \frac{d y}{d x} - y}{x ^{2}} : \frac{x \frac{d y}{d x} - y}{x y}

= \frac{x \frac{d y}{d x} - y}{x y}

Popular Examples

(\partial)/(\partial y)(x^2-2x-xy)

\frac{\partial}{\partial y} (x^{2} - 2 x - x y)

derivative of (x^2)/((x^2-1)^4)derivative of

\frac{x ^{2}}{( x ^{2} - 1 ) ^{4}}

(\partial)/(\partial y)(x^2e^y)

\frac{\partial}{\partial y} (x^{2} e^{y})

slope ofintercept (7)(-1.6)slope intercept

(7) (- 1.6)

x^2y^'+2xy=11cos^2(x)

x^{2} y^{^{'}} + 2 x y = 11 cos^{2} (x)

Frequently Asked Questions (FAQ)

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