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Chapter 3: Problem 13

Find the derivative of the following functions. $$g(w)=e^{w}\left(w^{3}-1\right)$$

Short Answer

Expert verified
Question: Find the derivative of the function $$g(w) = e^{w}(w^{3} - 1)$$. Answer: The derivative of the given function is $$g'(w) = e^{w}\left(w^{3} + 3w^{2} - 1\right)$$.

Step by step solution

01

Identify the functions and their derivatives

We have the function $$g(w) = e^{w}\left(w^{3}-1\right)$$, and we need to apply the product rule. Let's identify the two functions and find their derivatives: Function 1: $$f(w) = e^{w}$$ Derivative of Function 1: $$f'(w) = e^{w}$$ Function 2: $$g(w) = w^{3} - 1$$ Derivative of Function 2: $$g'(w) = 3w^{2}$$ (using the power rule)
02

Applying the product rule

Now that we have the derivatives for each function, we can apply the product rule to find the derivative of the original function: $$g'(w) = f'(w) \cdot g(w) + f(w) \cdot g'(w)$$ Substitute the function derivatives we found in Step 1: $$g'(w) = e^{w} \cdot (w^{3} - 1) + e^{w} \cdot 3w^{2}$$
03

Simplifying the expression

Now we want to simplify the expression for the derivative. We can factor out a common factor of $$e^{w}$$: $$g'(w) = e^{w}(w^{3} - 1 + 3w^{2})$$ This is the final simplified derivative of the original function: $$g'(w) = e^{w}\left(w^{3} + 3w^{2} - 1\right)$$

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Most popular questions from this chapter

Calculate the following derivatives using the Product Rule. $$\begin{array}{lll} \text { a. } \frac{d}{d x}\left(\sin ^{2} x\right) & \text { b. } \frac{d}{d x}\left(\sin ^{3} x\right) & \text { c. } \frac{d}{d x}\left(\sin ^{4} x\right) \end{array}$$ d. Based upon your answers to parts (a)-(c), make a conjecture about \(\frac{d}{d x}\left(\sin ^{n} x\right),\) where \(n\) is a positive integer. Then prove the result by induction.

Vertical tangent lines a. Determine the points at which the curve \(x+y^{3}-y=1\) has a vertical tangent line (see Exercise 52 ). b. Does the curve have any horizontal tangent lines? Explain.

Proof by induction: derivative of \(e^{k x}\) for positive integers \(k\) Proof by induction is a method in which one begins by showing that a statement, which involves positive integers, is true for a particular value (usually \(k=1\) ). In the second step, the statement is assumed to be true for \(k=n\), and the statement is proved for \(k=n+1,\) which concludes the proof. a. Show that \(\frac{d}{d x}\left(e^{k x}\right)=k e^{k x}\) for \(k=1\) b. Assume the rule is true for \(k=n\) (that is, assume \(\left.\frac{d}{d x}\left(e^{n x}\right)=n e^{n x}\right),\) and show this implies that the rule is true for \(k=n+1 .\) (Hint: Write \(e^{(n+1) x}\) as the product of two functions, and use the Product Rule.)

The hands of the clock in the tower of the Houses of Parliament in London are approximately \(3 \mathrm{m}\) and \(2.5 \mathrm{m}\) in length. How fast is the distance between the tips of the hands changing at 9:00? (Hint: Use the Law of cosines.)

A particle travels clockwise on a circular path of diameter \(R,\) monitored by a sensor on the circle at point \(P ;\) the other endpoint of the diameter on which the sensor lies is \(Q\) (see figure). Let \(\theta\) be the angle between the diameter \(P Q\) and the line from the sensor to the particle. Let \(c\) be the length of the chord from the particle's position to \(Q\) a. Calculate \(d \theta / d c\) b. Evaluate \(\left.\frac{d \theta}{d c}\right|_{c=0}\)
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