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Chapter 3: Problem 75

Calculate the derivative of the following functions (i) using the fact that \(b^{x}=e^{x \ln b}\) and (ii) by using logarithmic differentiation. Verify that both answers are the same. $$y=3^{x}$$

Short Answer

Expert verified
Answer: The derivative of the function \(y=3^x\) is \(\frac{dy}{dx} = 3^x \ln 3\) using both methods.

Step by step solution

01

Exponential Substitution

We're given the function \(y=3^x\). First, we will rewrite it using the exponential substitution method: \(b^x=e^{x\ln b}\). So, the function becomes: $$y=e^{x\ln 3}$$
02

Differentiate using the exponential substitution method

Now, we will differentiate the function with respect to \(x\): $$\frac{dy}{dx} = \frac{d}{dx}(e^{x\ln 3})$$ Using the chain rule for differentiation: $$\frac{dy}{dx} = e^{x\ln 3} \cdot \frac{d}{dx}(x\ln 3)$$ Since \(\ln 3\) is a constant: $$\frac{dy}{dx} = e^{x\ln 3}\cdot\ln 3 = 3^x \ln 3$$ We found the first derivative of \(y=3^x\) using the exponential substitution method.
03

Logarithmic Differentiation

Next, we will use the logarithmic differentiation method for the same function \(y=3^x\). Firstly, we take the natural logarithm of both sides of the equation: $$\ln y = \ln(3^x)$$ Using the properties of logarithms, we can simplify the expression: $$\ln y = x\ln 3$$
04

Differentiate using Logarithmic Differentiation

Now, we differentiate both sides of the equation with respect to \(x\): $$\frac{1}{y}\frac{dy}{dx} = \frac{d}{dx}(x\ln 3)$$ Since \(\ln 3\) is a constant: $$\frac{1}{y}\frac{dy}{dx} = \ln 3$$ Now, we multiply both sides by \(y\): $$\frac{dy}{dx} = y\ln 3$$ We substitute the original function, \(y=3^x\), into the equation: $$\frac{dy}{dx} = 3^x \ln 3$$ We found the first derivative of \(y=3^x\) using the logarithmic differentiation method.
05

Compare the Solutions

Now we compare both derivatives calculated in Step 2 and Step 4: $$3^x \ln 3 = 3^x \ln 3$$ Both answers are the same, so we have verified that both methods yield the same result for the derivative of the function \(y=3^x\): $$\frac{dy}{dx} = 3^x \ln 3$$

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Most popular questions from this chapter

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

A challenging second derivative Find \(\frac{d^{2} y}{d x^{2}},\) where \(\sqrt{y}+x y=1\).

Derivatives from tangent lines Suppose the line tangent to the graph of \(f\) at \(x=2\) is \(y=4 x+1\) and suppose \(y=3 x-2\) is the line tangent to the graph of \(g\) at \(x=2 .\) Find an equation of the line tangent to the following curves at \(x=2\) a. \(y=f(x) g(x)\) b. \(y=\frac{f(x)}{g(x)}\)

Derivatives and inverse functions $$\text { Find }\left(f^{-1}\right)^{\prime}(3) \text { if } f(x)=x^{3}+x+1$$

Proof of the Quotient Rule Let \(F=f / g\) be the quotient of two functions that are differentiable at \(x\) a. Use the definition of \(F^{\prime}\) to show that \(\frac{d}{d x}\left[\frac{f(x)}{g(x)}\right]=\lim _{h \rightarrow 0} \frac{f(x+h) g(x)-f(x) g(x+h)}{h g(x+h) g(x)}\) b. Now add \(-f(x) g(x)+f(x) g(x)\) (which equals 0) to the numerator in the preceding limit to obtain $$\lim _{h \rightarrow 0} \frac{f(x+h) g(x)-f(x) g(x)+f(x) g(x)-f(x) g(x+h)}{h g(x+h) g(x)}$$ Use this limit to obtain the Quotient Rule. c. Explain why \(F^{\prime}=(f / g)^{\prime}\) exists, whenever \(g(x) \neq 0\)
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