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Chapter 3: Problem 43

Compute the derivative of the following functions. $$f(x)=x e^{7 x}$$

Short Answer

Expert verified
Solution: The derivative of the function is \(f'(x) = e^{7x}(1 + 7x)\).

Step by step solution

01

Find the derivatives of u(x) and v(x)

To find the derivative of \(u(x) = x\), we simply take the derivative of x with respect to x: $$\frac{du}{dx} = 1$$ For the derivative of \(v(x) = e^{7x}\), we use the chain rule. The chain rule states that for a composite function \(g(h(x))\), the derivative is \(g'(h(x)) \cdot h'(x)\). In this case, our composite function is \(e^{7x}\), with \(g(x) = e^x\) and \(h(x) = 7x\). The derivative of \(g(x)\) is \(e^x\), and the derivative of \(h(x)\) is 7. Applying the chain rule, we get: $$\frac{dv}{dx} = e^{7x} \cdot 7$$
02

Apply the product rule

Now we have the derivatives of u and v, we can apply the product rule to find the derivative of f(x): $$\frac{df}{dx} = \frac{du}{dx} v(x) + u(x) \frac{dv}{dx}$$ Plugging in the derivatives we found in step 1: $$\frac{df}{dx} = 1 \cdot e^{7x} + x \cdot 7 e^{7x}$$
03

Simplify the expression

Finally, we can simplify the expression for the derivative: $$\frac{df}{dx} = e^{7x} + 7x e^{7x}$$ We can factor out \(e^{7x}\) from both terms: $$\frac{df}{dx} = e^{7x}(1 + 7x)$$ So the derivative of the given function is: $$f'(x) = e^{7x}(1 + 7x)$$

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