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Chapter 3: Problem 30

Derivatives of \(b^{x}\) Find the derivatives of the following functions. $$y=\ln 10^{x}$$

Short Answer

Expert verified
Answer: The derivative of the given function $$y=\ln 10^{x}$$ with respect to x is \(\ln 10\).

Step by step solution

01

Simplify the function using logarithmic identities

Recall the logarithm identity: $$\ln a^{x} = x \ln a$$ Applying this to our function, we get $$y = x \ln 10$$
02

Apply the Chain Rule

Now that we have simplified the function, we can apply the chain rule to find the derivative of y with respect to x: $$\frac{dy}{dx} = \frac{d}{dx} (x \ln 10)$$
03

Find the derivative

Since \(\ln 10\) is a constant, differentiating \(x \ln 10\) with respect to x gives: $$\frac{dy}{dx} = \ln 10 \cdot \frac{d}{dx}(x)$$ And the derivative of x with respect to x is 1, so the final result is: $$\frac{dy}{dx} = \ln 10$$ Thus, the derivative of the given function $$y=\ln 10^{x}$$ with respect to x is \(\ln 10\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithmic Identities
Understanding logarithmic identities is crucial when simplifying expressions before taking derivatives. Logarithms have properties that can transform complex expressions into more manageable ones. For example, the identity \(\ln a^x = x \ln a\) allows us to take an exponent and turn it into a multiplication, which is far easier to differentiate.When you encounter a function like \(y = \ln 10^x\), remember that according to the logarithmic identity, you can simplify it to \(y = x \ln 10\). This step is vital because it converts the problem into a form that allows for straightforward application of differentiation rules.
Chain Rule in Calculus
The chain rule is a fundamental differentiation technique in calculus. It's used when you need to find the derivative of a composite function - a function composed of two or more functions. The general form of the chain rule is expressed as:\[\frac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x)\]In the context of logarithmic functions, the chain rule often simplifies the differentiation process. After applying logarithmic identities, like in our exercise, the chain rule might look like it's not necessary. However, having a deep understanding of when and how to apply the chain rule will help you tackle more complex functions that cannot be simplified using logarithmic identities alone.
Differentiation Techniques
There are several techniques one can apply when differentiating functions. In the exercise we're discussing, after using logarithmic identities to simplify the function, we proceed with direct differentiation. Here, differentiation techniques are relatively straightforward. Since \(\ln 10\) is a constant, and the derivative of \(x\) with respect to \(x\) is 1, we can multiply these to find the derivative of the function.It's important also to recognize when different functions require different techniques. For instance, in cases where the function鈥檚 complexity does not allow simplification, you may need to apply the product rule, quotient rule, or implicit differentiation. Always assess the function to determine the most efficient and appropriate method for finding the derivative.

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Most popular questions from this chapter

Work carefully Proceed with caution when using implicit differentiation to find points at which a curve has a specified slope. For the following curves, find the points on the curve (if they exist) at which the tangent line is horizontal or vertical. Once you have found possible points, make sure they actually lie on the curve. Confirm your results with a graph. $$x^{2}(y-2)-e^{y}=0$$

Prove the following identities and give the values of \(x\) for which they are true. $$\tan \left(2 \tan ^{-1} x\right)=\frac{2 x}{1-x^{2}}$$

Tangency question It is easily verified that the graphs of \(y=x^{2}\) and \(y=e^{x}\) have no points of intersection (for \(x>0\) ), and the graphs of \(y=x^{3}\) and \(y=e^{x}\) have two points of intersection. It follows that for some real number \(2 0 \) ). Using analytical and/or graphical methods, determine \(p\) and the coordinates of the single point of intersection.

The flow of a small stream is monitored for 90 days between May 1 and August 1. The total water that flows past a gauging station is given by $$V(t)=\left\\{\begin{array}{ll}\frac{4}{5} t^{2} & \text { if } 0 \leq t<45 \\\\-\frac{4}{5}\left(t^{2}-180 t+4050\right) & \text { if } 45 \leq t<90, \end{array}\right.$$ where \(V\) is measured in cubic feet and \(t\) is measured in days, with \(t=0\) corresponding to May 1. a. Graph the volume function. b. Find the flow rate function \(V^{\prime}(t)\) and graph it. What are the units of the flow rate? c. Describe the flow of the stream over the 3 -month period. Specifically, when is the flow rate a maximum?

An observer is \(20 \mathrm{m}\) above the ground floor of a large hotel atrium looking at a glass-enclosed elevator shaft that is \(20 \mathrm{m}\) horizontally from the observer (see figure). The angle of elevation of the elevator is the angle that the observer's line of sight makes with the horizontal (it may be positive or negative). Assuming that the elevator rises at a rate of \(5 \mathrm{m} / \mathrm{s}\), what is the rate of change of the angle of elevation when the elevator is \(10 \mathrm{m}\) above the ground? When the elevator is \(40 \mathrm{m}\) above the ground?
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