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Chapter 3: Problem 12

Find the derivative of the following functions. $$f(x)=\left(1+\frac{1}{x^{2}}\right)\left(x^{2}+1\right)$$

Short Answer

Expert verified
The derivative of the function is $$f'(x) = 2x - \frac{2}{x^3}$$

Step by step solution

01

Find the derivative of u(x)

To find the derivative of $$u(x) = \left(1+\frac{1}{x^{2}}\right)$$, we need to apply the power rule to the second term. The power rule states that if $$g(x) = x^n$$, then $$g'(x) = nx^{n-1}$$. In this case, we have the term $$\frac{1}{x^2}$$ which can be written as $$x^{-2}$$. Applying the power rule, we get: $$\frac{d}{dx}\left(1+\frac{1}{x^{2}}\right) = 0-2x^{-3} = -\frac{2}{x^3}$$
02

Find the derivative of v(x)

To find the derivative of $$v(x) = \left(x^{2}+1\right)$$, we need to apply the power rule on the first term. Applying the power rule, we get: $$\frac{d}{dx}\left(x^{2}+1\right) = 2x+0 = 2x$$
03

Apply the product rule

Now we can apply the product rule to find the derivative of the entire function, using the derivatives found in Steps 1 and 2: The derivative of the given function will be: $$f'(x) = u'(x)v(x) + u(x)v'(x)$$ $$f'(x) = \left(-\frac{2}{x^3}\right)\left(x^{2}+1\right) + \left(1+\frac{1}{x^{2}}\right)\left(2x\right)$$
04

Simplify the answer

By simplifying the resulting expression, we can combine terms to make the answer more manageable: $$f'(x) = -\frac{2}{x} - \frac{2}{x^3} + 2x + \frac{2}{x}$$ $$f'(x) = 2x - \frac{2}{x^3}$$ So the derivative of the given function is: $$f'(x) = 2x - \frac{2}{x^3}.$$

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Most popular questions from this chapter

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