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Chapter 3: Problem 45

Compute the derivative of the following functions. $$f(x)=15 e^{3 x}$$

Short Answer

Expert verified
Answer: The derivative of the function \(f(x)=15e^{3x}\) is \(f'(x) = 45e^{3x}\).

Step by step solution

01

Find the derivative of \(g(x)\), \(g'(x)\)

First, let's find the derivative of \(g(x)=3x\) with respect to \(x\). The derivative of a linear function is simply its slope. $$g'(x) = \frac{d}{dx}(3x) = 3$$ So, \(g'(x)=3\).
02

Find the derivative of \(f(u)\), \(f'(u)\)

Now, let's find the derivative of \(f(u)=15e^u\) with respect to \(u\). The derivative of an exponential function with base \(e\) is equal to the original function. $$f'(u) = \frac{d}{du}(15e^u) = 15e^u$$ So, \(f'(u)=15e^u\).
03

Apply the chain rule to find the derivative of \(f(x)\)

Now we can use the chain rule to find the derivative of the original function \(f(x)=15e^{3x}\). Recall that the chain rule states: $$(f(g(x)))' = f'(g(x)) \cdot g'(x)$$ First, let's find \(f'(g(x))\). We computed that \(f'(u)=15e^u\). So, we plug in \(g(x)=3x\) for \(u\): $$f'(g(x)) = 15e^{3x}$$ Therefore, the derivative of \(f(x)\) is: $$f'(x) = f'(g(x)) \cdot g'(x) = (15e^{3x})(3) = 45e^{3x}$$ The derivative of the function \(f(x)=15e^{3x}\) is \(f'(x)=45e^{3x}\).

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Most popular questions from this chapter

Use the following table to find the given derivatives. $$\begin{array}{llllll} x & 1 & 2 & 3 & 4 & 5 \\ \hline f(x) & 5 & 4 & 3 & 2 & 1 \\ f^{\prime}(x) & 3 & 5 & 2 & 1 & 4 \\ g(x) & 4 & 2 & 5 & 3 & 1 \\ g^{\prime}(x) & 2 & 4 & 3 & 1 & 5 \end{array}$$ $$\left.\frac{d}{d x}(x f(x))\right|_{x=3}$$

A lighthouse stands 500 m off of a straight shore, the focused beam of its light revolving four times each minute. As shown in the figure, \(P\) is the point on shore closest to the lighthouse and \(Q\) is a point on the shore 200 m from \(P\). What is the speed of the beam along the shore when it strikes the point \(Q ?\) Describe how the speed of the beam along the shore varies with the distance between \(P\) and \(Q\). Neglect the height of the lighthouse.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The derivative \(\frac{d}{d x}\left(e^{5}\right)\) equals \(5 \cdot e^{4}\) b. The Quotient Rule must be used to evaluate \(\frac{d}{d x}\left(\frac{x^{2}+3 x+2}{x}\right)\) c. \(\frac{d}{d x}\left(\frac{1}{x^{5}}\right)=\frac{1}{5 x^{4}}\) d. \(\frac{d^{n}}{d x^{n}}\left(e^{3 x}\right)=3^{n} \cdot e^{3 x},\) for any integer \(n \geq 1\)

Use the following table to find the given derivatives. $$\begin{array}{llllll} x & 1 & 2 & 3 & 4 & 5 \\ \hline f(x) & 5 & 4 & 3 & 2 & 1 \\ f^{\prime}(x) & 3 & 5 & 2 & 1 & 4 \\ g(x) & 4 & 2 & 5 & 3 & 1 \\ g^{\prime}(x) & 2 & 4 & 3 & 1 & 5 \end{array}$$ $$\left.\frac{d}{d x}\left[\frac{x f(x)}{g(x)}\right]\right|_{x=4}$$

Use any method to evaluate the derivative of the following functions. \(y=\frac{x-a}{\sqrt{x}-\sqrt{a}} ; a\) is a positive constant.
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