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Chapter 2: Problem 73

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. If \(f\) and \(g\) have second derivatives, then $$\frac{d}{d x}\left[f(x) g^{\prime}(x)-f^{\prime}(x) g(x)\right]=f(x) g^{\prime \prime}(x)-f^{\prime \prime}(x) g(x)$$

Short Answer

Expert verified
The statement is true. The derivative of the expression $$f(x) g^{\prime}(x)-f^{\prime}(x) g(x)$$ with respect to \(x\) is indeed $$f(x)g^{\prime \prime}(x)-f^{\prime \prime}(x) g(x)$$, as confirmed by applying the product rule and combining the terms.

Step by step solution

01

Differentiate the given expression

We are given the expression $$f(x) g^{\prime}(x)-f^{\prime}(x) g(x)$$ and our task is to find its derivative with respect to \(x\). We'll first differentiate this expression step-by-step and then simplify it.
02

Apply the product rule

Recall that the product rule states that if \(u(x)\) and \(v(x)\) are functions of \(x\) with derivatives, then the derivative of their product is $$\frac{d}{dx}(u(x)v(x)) = u'(x)v(x) + u(x)v'(x)$$ We will apply the product rule to both terms in our expression separately. Let \(u_1(x) = f(x)\) and \(v_1(x) = g^{\prime}(x)\). Then we have: $$\frac{d}{dx} [u_1(x) v_1(x)] = f'(x)g'(x) + f(x)g''(x)$$ Now, let \(u_2(x) = f^{\prime}(x)\) and \(v_2(x) = g(x)\): $$\frac{d}{dx} [u_2(x) v_2(x)] = f''(x) g(x) + f'(x) g'(x)$$
03

Combine the terms

Now we will combine the derivatives we found for the two terms in our expression: $$\frac{d}{dx}\left[f(x) g^{\prime}(x)-f^{\prime}(x) g(x)\right]=f'(x)g'(x) + f(x)g''(x) - f''(x) g(x) - f'(x) g'(x)$$
04

Check if the expression matches the given statement

We will now check if the result of our differentiation matches the given statement: $$f(x)g^{\prime \prime}(x)-f^{\prime \prime}(x) g(x)$$ We can see that it does match the given statement: $$\frac{d}{dx}\left[f(x) g^{\prime}(x)-f^{\prime}(x) g(x)\right]=f'(x)g'(x) + f(x)g''(x) - f''(x) g(x) - f'(x) g'(x) = f(x)g^{\prime \prime}(x)-f^{\prime \prime}(x) g(x)$$ Thus, the statement is true.

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Most popular questions from this chapter

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