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Chapter 7: Problem 19

Find the derivative of the function. $$ f(x)=4 \sqrt{x} $$

Short Answer

Expert verified
The derivative of the function \(f(x) = 4 \sqrt{x}\) is \(f'(x) = \frac{2}{\sqrt{x}}\).

Step by step solution

01

Rewrite the function

Rewrite the given function \(f(x) = 4 \sqrt{x}\) in the form of \(f(x) = ax^n\). Therefore, the function becomes \(f(x) = 4x^{1/2}\).
02

Apply the power rule

Use the power rule, which states that \(d/dx[x^n] = nx^{n-1}\) to find the derivative. The rule is applied to every term of the function. Here \(a = 4, n = \frac{1}{2}\). Applying the power rule to our function will result \(f'(x) = \frac{1}{2} \cdot 4x^{(\frac{1}{2}-1)}\).
03

Simplify the result

Simplify the expression to get the final answer, \(f'(x) = 2x^{-\frac{1}{2}}\). But remembering - a negative exponent means that the term is on the denominator. Therefore, the function can also be written as \(f'(x) = \frac{2}{x^\frac{1}{2}}\). This means for any x in the domain of \(f\), the derivative of the function \(f\) at \(x\) is \(f'(x) = \frac{2}{\sqrt{x}}\).

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Most popular questions from this chapter

Find \(d y / d u, d u / d x\), and \(d y / d x\). $$ y=u^{3}, u=3 x^{2}-2 $$

Use the General Power Rule to find the derivative of the function. $$ y=(2 x-7)^{3} $$

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ y=t \sqrt{t+1} $$

Identify the inside function, \(u=g(x)\), and the outside function, \(y=f(u)\). $$ y=f(g(x)) \quad u=g(x) \quad y=f(u) $$ $$ y=\left(x^{2}+1\right)^{4 / 3} $$

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ y=x \sqrt{2 x+3} $$
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