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Chapter 3: Problem 22

Find \(d y / d x\) for the following functions. $$y=e^{6 x} \sin x$$

Short Answer

Expert verified
Answer: The derivative of the function \(y=e^{6x}\sin x\) with respect to \(x\) is \(\frac{dy}{dx} = 6e^{6x}\sin x + e^{6x}\cos x\).

Step by step solution

01

Find the derivatives of f and g

We need to find the derivatives of \(f(x) = e^{6x}\) and \(g(x) = \sin x\) with respect to \(x\). For \(f(x) = e^{6x}\), use the chain rule, which says that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. So, taking the derivative with respect to \(x\), we have: $$f'(x) = \frac{d}{dx} (e^{6x}) = e^{6x} \cdot \frac{d(6x)}{dx} = e^{6x} \cdot 6 = 6e^{6x}$$ For \(g(x) = \sin x\), the derivative with respect to \(x\) is simply: $$g'(x) = \frac{d}{dx} (\sin x) = \cos x$$ Now, we can use the product rule to find the derivative of \(y = e^{6x}\sin x\).
02

Apply the product rule

Using the product rule, we get: $$\frac{dy}{dx} = f'(x)g(x) + f(x)g'(x) = 6e^{6x}\sin x + e^{6x}\cos x$$ So, the derivative of \(y=e^{6x}\sin x\) with respect to \(x\) is: $$\frac{dy}{dx} = 6e^{6x}\sin x + e^{6x}\cos x$$

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Most popular questions from this chapter

Visualizing tangent and normal lines a. Determine an equation of the tangent line and normal line at the given point \(\left(x_{0}, y_{0}\right)\) on the following curves. (See instructions for Exercises \(63-68 .)\) b. Graph the tangent and normal lines on the given graph. \(\left(x^{2}+y^{2}\right)^{2}=\frac{25}{3}\left(x^{2}-y^{2}\right);\) \(\left(x_{0}, y_{0}\right)=(2,-1)\) (lemniscate of Bernoulli)

Witch of Agnesi Let \(y\left(x^{2}+4\right)=8\) (see figure). a. Use implicit differentiation to find \(\frac{d y}{d x}\) b. Find equations of all lines tangent to the curve \(y\left(x^{2}+4\right)=8\) when \(y=1\) c. Solve the equation \(y\left(x^{2}+4\right)=8\) for \(y\) to find an explicit expression for \(y\) and then calculate \(\frac{d y}{d x}\) d. Verify that the results of parts (a) and (c) are consistent.

The following limits equal the derivative of a function \(f\) at a point a. a. Find one possible \(f\) and \(a\) b. Evaluate the limit. $$\lim _{x \rightarrow \pi / 4} \frac{\cot x-1}{x-\frac{\pi}{4}}$$

An observer stands \(20 \mathrm{m}\) from the bottom of a 10 -m-tall Ferris wheel on a line that is perpendicular to the face of the Ferris wheel. The wheel revolves at a rate of \(\pi\) rad/min and the observer's line of sight with a specific seat on the wheel makes an angle \(\theta\) with the ground (see figure). Forty seconds after that seat leaves the lowest point on the wheel, what is the rate of change of \(\theta ?\) Assume the observer's eyes are level with the bottom of the wheel.

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 \(n \geq 1\)
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