Chapter 2: Q. 72 (page 223)

Use implicit differentiation and the fact that $\log_{b} x$ is the inverse of $b^{x}$ to prove that $\frac{d}{d x} (\log_{b} x) = \frac{1}{x \ln b}$

Short Answer

Expert verified

The given differentiation has been proved.

Step by step solution

Step 1. Given Information

The given function is $\log_{b} x$

Step 2. Proof

Let $\log_{b} x = y \Rightarrow x = b^{y}$

Take log on both the sides, we get,

$\ln x = \ln b^{y} \ln x = y \ln b \frac{1}{x} = \ln b \frac{d y}{d x} \frac{d y}{d x} = \frac{1}{x \ln b} \frac{d}{d x} (\log_{b} x) = \frac{1}{x l n b}$

Hence, Proved.

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Use implicit differentiation and the fact that $\log_{b} x$ is the inverse of $b^{x}$ to prove that $\frac{d}{d x} (\log_{b} x) = \frac{1}{x \ln b}$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Proof

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Use implicit differentiation and the fact that logbx is the inverse of bx to prove thatddxlogbx=1xlnb

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Use implicit differentiation and the fact that $\log_{b} x$ is the inverse of $b^{x}$ to prove that $\frac{d}{d x} (\log_{b} x) = \frac{1}{x \ln b}$