Logarithmic Differentiation

If differentiation of a function is performed after taking logarithm on both sides, the process is called logarithmic differentiation.

• Step 1: Take the natural logarithm (\[\log_e\] or \[\ln\]) on both sides to bring the exponent down:

  \[\log y = v(x) \cdot \log[u(x)]\]

• Step 2: Differentiate both sides with respect to x using the chain rule and product rule:

  \[\frac{1}{y} \cdot \frac{dy}{dx} = v(x) \cdot \frac{d}{dx}(\log[u(x)]) + \log[u(x)] \cdot \frac{d}{dx}(v(x))\]

• Step 3: Isolate \[\frac{dy}{dx}\] by multiplying the entire right side by y:

  \[\frac{dy}{dx} = y \left[ \frac{v(x)}{u(x)} \cdot u'(x) + v'(x) \cdot \log[u(x)] \right]\]

Differentiate \[\sqrt{\frac{(x - 3)(x^2 + 4)}{3x^2 + 4x + 5}}\] w.r.t. \[x\].

Solution: Let \[y = \sqrt{\frac{(x - 3)(x^2 + 4)}{(3x^2 + 4x + 5)}}\]

Taking the logarithm on both sides, we have

Now, differentiating both sides w.r.t. \[x\], we get

or \[\frac{dy}{dx} = \frac{y}{2} \left[ \frac{1}{(x - 3)} + \frac{2x}{x^2 + 4} - \frac{6x + 4}{3x^2 + 4x + 5} \right]\]

\[= \frac{1}{2} \sqrt{\frac{(x - 3)(x^2 + 4)}{3x^2 + 4x + 5}} \left[ \frac{1}{(x - 3)} + \frac{2x}{x^2 + 4} - \frac{6x + 4}{3x^2 + 4x + 5} \right]\]

Differentiate \[x^{\sin x}, x > 0\] w.r.t. \[x\].

Solution Let \[y = x^{\sin x}\]. Taking the logarithm on both sides, we have

Therefore \[\frac{1}{y} \cdot \frac{dy}{dx} = \sin x \frac{d}{dx} (\log x) + \log x \frac{d}{dx} (\sin x)\]

\[\frac{1}{y} \frac{dy}{dx} = (\sin x) \frac{1}{x} + \log x \cos x\]

or \[\frac{dy}{dx} = y \left[ \frac{\sin x}{x} + \cos x \log x \right]\]

 \[= x^{\sin x} \left[ \frac{\sin x}{x} + \cos x \log x \right]\]

\[= x^{\sin x - 1} \cdot \sin x + x^{\sin x} \cdot \cos x \log x\]

• Use logarithmic differentiation when the function is a complex product, quotient, or variable exponent form.

• Write y = function first, then take \[\ln\] on both sides.

• Apply logarithmic rules before differentiating.

• Differentiate \[\ln y\] carefully: \[\frac{d}{dx}(\ln y) = \frac{1}{y} \frac{dy}{dx}\].

• Substitute the original value of y at the end.

• Ensure the expression inside logarithm remains positive.