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Chapter 3: Problem 67

Use any method to evaluate the derivative of the following functions. $$f(x)=4 x^{2}-\frac{2 x}{5 x+1}$$

Short Answer

Expert verified
Answer: The derivative of the function is \(f'(x) = 8x + \frac{2}{(5x+1)^2}\).

Step by step solution

01

Find the derivative of the quadratic term

Recall the power rule for differentiation, which states that for any function \(f(x) = ax^n\), the derivative is \(f'(x) = nax^{n-1}\). Apply the power rule on the quadratic term, \(4x^{2}\). The derivative is: $$\frac{d}{dx}(4x^{2}) = 2 \cdot 4x^{2-1} = 8x$$
02

Find the derivative of the fraction term

We need to find the derivative of \(\frac{2x}{5x+1}\). We will use the quotient rule for this term, which is given as follows for a function in the form \(f(x) = \frac{u(x)}{v(x)}\): $$f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{v^2(x)}$$ Here, \(u(x) = 2x\) and \(v(x) = 5x+1\). First, we need to find \(u'(x)\) and \(v'(x)\) using the power rule: $$u'(x) = \frac{d}{dx}(2x) = 2$$ $$v'(x) = \frac{d}{dx}(5x+1) = 5$$ Now, use the quotient rule to find the derivative of the fraction term: $$\frac{d}{dx}\left(\frac{2x}{5x+1}\right) = \frac{2(5x+1) - 2x(5)}{(5x+1)^2} = \frac{2}{(5x+1)^2}$$
03

Combine the derivatives

To find the overall derivative of the function, we need to combine the derivatives of the two terms we found in Step 1 and Step 2: $$f'(x) = \frac{d}{dx}(4x^{2}) + \frac{d}{dx}\left(\frac{2x}{5x+1}\right)$$ So, the derivative of the given function is: $$f'(x) = 8x + \frac{2}{(5x+1)^2}$$

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Most popular questions from this chapter

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