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Chapter 3: Problem 46

Compute the derivative of the following functions. $$y=3 x^{2}-2 x+e^{-2 x}$$

Short Answer

Expert verified
Question: Find the derivative of the function \(y=3x^2-2x+e^{-2x}\). Answer: The derivative of the function is \(\frac{dy}{dx} = 6x - 2 - 2e^{-2x}\).

Step by step solution

01

Identify the components of the function

The given function can be seen as the combination of three functions: $$y=3x^2$$ $$y=-2x$$ $$y=e^{-2x}$$
02

Differentiate each component separately

Differentiate the first component function \(3x^2\) using the power rule. The power rule states that if \(y=cx^n\), then the derivative \(\frac{dy}{dx}=cnx^{n-1}\). $$\frac{d}{dx}(3x^2) = 6x^{2-1}=6x$$ Differentiate the second component function \(-2x\). As this is a linear function, its derivative is simply the slope of the line. $$\frac{d}{dx}(-2x)=-2$$ Differentiate the third component function \(e^{-2x}\) using the chain rule. The chain rule states that if \(y=f(g(x))\), then \(\frac{dy}{dx}=f'(g(x))g'(x)\). In this case, \(f(u)=e^u\) and \(g(x)=-2x\). $$\frac{d}{dx}(e^{-2x})=(e^u)'(-2)=-2e^{-2x}$$
03

Add the derivatives together

Now that we have found the derivatives of the three individual functions, we can add them together to get the derivative of the entire function. $$\frac{dy}{dx} = 6x - 2 - 2e^{-2x}$$ So, the derivative of the given function is: $$\frac{dy}{dx} = 6x - 2 - 2e^{-2x}$$

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Most popular questions from this chapter

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