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Chapter 3: Problem 56

Calculate the derivative of the following functions. $$y=\log _{10} x$$

Short Answer

Expert verified
Answer: The derivative of the function $$y=\log _{10} x$$ is $$\frac{dy}{dx} = \frac{1}{x \ln{10}}$$.

Step by step solution

01

Convert to natural logarithm

We will use the logarithmic property $$\log _{a} b = \frac{\ln{b}}{\ln{a}}$$ to convert the given function into natural logarithm. By using this property, we get: $$y=\frac{\ln{x}}{\ln{10}}$$
02

Apply the chain rule

Now that we have an expression involving the natural logarithm, we can apply the chain rule as follows: $$\frac{dy}{dx} = \frac{d}{dx} \left( \frac{\ln{x}}{\ln{10}} \right) = \frac{1}{\ln{10}} \cdot \frac{d}{dx} (\ln{x})$$
03

Calculate the derivative

Now we calculate the derivative of the natural logarithm of x: $$\frac{d}{dx} (\ln{x}) = \frac{1}{x}$$
04

Multiply by the constant

Finally, we multiply the result from Step 3 by $$\frac{1}{\ln{10}}$$ to get the derivative of the given function: $$\frac{dy}{dx} = \frac{1}{\ln{10}} \cdot \frac{1}{x} = \frac{1}{x \ln{10}}$$ The derivative of the function $$y=\log _{10} x$$ is $$\frac{dy}{dx} = \frac{1}{x \ln{10}}$$.

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Most popular questions from this chapter

Continuity of a piecewise function Let $$f(x)=\left\\{\begin{aligned} \frac{3 \sin x}{x} & \text { if } x \neq 0 \\ a\ \ \ \ \ & \text { if } x=0 \end{aligned}\right.$$ For what values of \(a\) is \(f\) continuous?

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Quotient Rule for the second derivative Assuming the first and second derivatives of \(f\) and \(g\) exist at \(x\), find a formula for \(\frac{d^{2}}{d x^{2}}\left[\frac{f(x)}{g(x)}\right]\)

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