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Chapter 3: Problem 8

Find the derivative of the following functions. $$g(x)=6 x-2 x e^{x}$$

Short Answer

Expert verified
Answer: The derivative of the function is given by \(g'(x) = 6 - 2e^x - 2xe^x\).

Step by step solution

01

Differentiate the first term

We will differentiate the first term first \((6x)\). The power rule states that to differentiate \(x^n\), we multiply the constant coefficient by the exponent and subtract one from the exponent. Applying the power rule on \(6x\), we get: $$\frac{d}{dx}(6x) = 6\cdot x^{1-1} = 6x^0 = 6$$
02

Differentiate the second term using the product rule

Now we need to find the derivative of the second term \(-2xe^{x}\). Since this term is a product of two functions, we'll use the product rule. The product rule states that \((uv)'=u'v+uv'\), where u(x) = -2x and v(x) = \(e^x\). First, we differentiate each function individually: $$u(x) = -2x$$ $$\frac{d}{dx}(-2x) = -2$$ $$v(x) = e^x$$ $$\frac{d}{dx}(e^x) = e^x$$
03

Apply the product rule to the second term

Now we apply the product rule formula \((uv)'=u'v+uv'\): $$(-2xe^{x})' = (-2)(e^x) + (-2x)(e^x)$$ By simplifying, we get: $$-2e^x - 2xe^x$$
04

Combine the derivatives of both terms and write the final answer

Finally, we need to add the derivative of the first term and the derivative of the second term: $$g'(x) = \frac{d}{dx}(6x-2xe^x) = 6 - 2e^x - 2xe^x$$ So, the derivative of the given function is: $$g'(x) = 6 - 2e^x - 2xe^x$$

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