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Chapter 3: Problem 41

Use the General Power Rule where appropriate to find the derivative of the following functions. $$f(x)=(2 x-3) x^{3 / 2}$$

Short Answer

Expert verified
Answer: The derivative of the function is $$f'(x) = 5x^{3/2} - \frac{9}{2}x^{1/2}$$.

Step by step solution

01

Identify the function

We are given the following function to find its derivative: $$f(x) = (2x - 3)x^{3/2}$$
02

Applying General Power Rule

According to the General Power Rule for derivatives, the derivative of a function multiplied by a power of x (in our case, \(x^{3/2}\)) is: $$\frac{d}{dx}(u(x) \cdot v(x)) = u'(x) \cdot v(x) + u(x) \cdot v'(x)$$ where \(u(x)\) and \(v(x)\) are differentiable functions. In our case, we have \(u(x) = 2x - 3\) and \(v(x) = x^{3/2}\), and we need to find their derivatives.
03

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

To find the derivative of u(x) and v(x), we will use the basic power rule for derivatives: $$\frac{d}{dx}(cx^n) = n \cdot cx^{n-1}$$ For \(u(x) = 2x - 3\), the derivatives are: $$u'(x) = \frac{d}{dx}(2x) - \frac{d}{dx}(3) = 2 - 0 = 2$$ For \(v(x) = x^{3/2}\), the derivatives are: $$v'(x) = \frac{3}{2}x^{(3/2)-1} = \frac{3}{2}x^{1/2}$$
04

Apply the General Power Rule to find the derivative of f(x)

Now that we have the derivatives of u(x) and v(x), we can apply the General Power Rule to find the derivative of f(x): $$\frac{d}{dx}(f(x)) = u'(x) \cdot v(x) + u(x) \cdot v'(x) = 2 \cdot x^{3/2} + (2x-3) \cdot \frac{3}{2}x^{1/2}$$
05

Simplify the final expression

Now, we simplify the expression for the derivative of f(x): $$\frac{d}{dx}(f(x)) = 2x^{3/2} + 3x^{3/2} - \frac{9}{2}x^{1/2}$$ So, the derivative of the function is: $$f'(x) = \frac{d}{dx}(f(x)) = 5x^{3/2} - \frac{9}{2}x^{1/2}$$

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