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Chapter 2: Problem 9

Find the derivative with respect to \(x\) of the function y specified implicitly by \(\int_{0}^{y} e^{t} d t+\int_{0}^{x} \cos t d t=0\)

Short Answer

Expert verified
Question: Find the derivative of the function y with respect to x, given the equation \(\int_{0}^{y} e^{t} dt + \int_{0}^{x} \cos t dt = 0\). Answer: \(\frac{dy}{dx} = \frac{-\cos{x}}{e^{y}}\)

Step by step solution

01

Differentiate both sides with respect to x

Using the chain rule for differentiation, we differentiate the given equation: \(\frac{d}{dx} \left(\int_{0}^{y} e^{t} dt\right) + \frac{d}{dx} \left(\int_{0}^{x} \cos t dt\right) = \frac{d}{dx} (0)\)
02

Apply the Fundamental Theorem of Calculus to differentiate the integral expressions

We have: \(\frac{d}{dx} \left(\int_{0}^{y} e^{t} dt\right) = e^{y} \frac{dy}{dx}\) and \(\frac{d}{dx} \left(\int_{0}^{x} \cos t dt\right) = \cos{x}\) So, the differentiated equation becomes: \(e^{y}\frac{dy}{dx}+\cos{x} = 0\)
03

Solve for \(\frac{dy}{dx}\)

Lastly, we isolate \(\frac{dy}{dx}\): \(\frac{dy}{dx}=\frac{-\cos{x}}{e^{y}}\) The derivative of the function y with respect to x is: \(\frac{dy}{dx}=\frac{-\cos{x}}{e^{y}}\)

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Most popular questions from this chapter

Evaluate the following integrals: (i) \(\int_{-\infty}^{\infty} \frac{x d x}{x^{4}+1}\) (ii) \(\int_{0}^{1} \frac{\ln (1-x)}{x} \mathrm{dx}\) (iii) \(\int_{0}^{\infty} \frac{\mathrm{dx}}{(\mathrm{x}+1)(\mathrm{x}+2)}\) (iv) \(\int_{0}^{\infty} \frac{x^{2} d x}{\left(x^{2}+a^{2}\right)\left(x^{2}+b^{2}\right)}, a, b>0 .\)

A function \(\mathrm{f}\), continuous on the positive real axis, has the property that \(\int_{1}^{x y} f(t) d t=y \int_{1}^{x} f(t) d t+x \int_{1}^{y} f(t) d t\) for all \(x>0\) and all \(y>0 .\) If \(f(1)=3\), compute \(\mathrm{f}(\mathrm{x})\) for each \(\mathrm{x}>0\).

Assume \(\int\) is continuous on \([a, b]\). Assume also that \(\int_{a}^{b} f(x) g(x) d x=0\) for every function \(g\) that is continuous on \([\mathrm{a}, \mathrm{b}]\). Prove that \(\mathrm{f}(\mathrm{x})=0\) for all xin [a. b]

Suppose that the velocity function of a particle moving along a line is \(v(t)=3 t^{3}+2\). Find the average velocity of the particle over the time interval \(1 \leq \mathrm{t} \leq 4\) by integrating.

Evaluate \(\int_{0}^{1} \frac{\tan ^{-1} \mathrm{ax}}{\mathrm{x} \sqrt{1-\mathrm{x}^{2}}} \mathrm{dx}\)
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