Question

Verify the statement by showing that the derivative of the right side is equal to the integrand of the left side. $$ \int \frac{4}{\sqrt{x}} d x=8 \sqrt{x}+C $$

Short answer

After differentiating the right side of the equation, which is \(8 \sqrt{x}+C\), with respect to x using the power rule, the derived expression is \( \frac{4}{\sqrt{x}}\), matching the original integrand of the left side of the equation. Therefore the verification is successful because the derivative of the right side equals the integrand of the left side.

Step-by-step solution

Identify the Function to Differentiate

The first step is to identify the function that needs to be differentiated. In this case, the right side of the equation, which is \(8 \sqrt{x}+C\), is what needs to be differentiated.

Apply the Power Rule for Differentiation

Next step is to differentiate \(8 \sqrt{x}+C\) with respect to x. When you differentiate \(8 \sqrt{x}+C\) with respect to x, use the power rule, which states that if \(f(x) = x^n\), then \(f'(x) = n*x^{n-1}\). Applying the power rule to this function, notice that \(\sqrt{x}\) is the same as \(x^{1/2}\). The derivative of a constant (C) is zero.

Compare the Differentiated Function with the Original Integrand

In step 3, the result from the derivative should be compared to the original integrand of the left side of the given expression. The original integrand is \( \frac{4}{\sqrt{x}}\). Re-writing the integral in the form of a power of x makes comparing easier, i.e., \( \frac{4}{x^{1/2}}\).

Simplify and Confirm Equality

Simplify the expressions and determine if they are equal. If they are identical, then the verification is successful