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Chapter 3: Problem 12

Use Version I of the Chain Rule to calculate \(\frac{d y}{d x}\). $$y=\sqrt{7 x-1}$$

Short Answer

Expert verified
Answer: The derivative of the function \(y=\sqrt{7x-1}\) with respect to \(x\) is \(\frac{dy}{dx}=\frac{7}{2\sqrt{7x-1}}\).

Step by step solution

01

Identify the outer and inner functions

In this problem, the function given is \(y=\sqrt{7x-1}\). The outer function is \(f(u)=\sqrt{u}\) and the inner function is \(g(x)=7x-1\).
02

Differentiate the outer function

To differentiate the outer function \(f(u)=\sqrt{u}\) with respect to \(u\), recall the power rule for differentiation: \(\frac{d}{du}(u^n)=nu^{n-1}\). Applying this rule, we get: $$\frac{df}{du}=\frac{1}{2\sqrt{u}}$$
03

Differentiate the inner function

Now we differentiate the inner function \(g(x)=7x-1\) with respect to \(x\). Using the power rule for differentiation again, we get: $$\frac{dg}{dx}=7$$
04

Apply the chain rule

The chain rule states that \(\frac{dy}{dx}=\frac{df}{du}\cdot\frac{dg}{dx}\), where \(u=g(x)\). In our problem, we have \(\frac{df}{du}=\frac{1}{2\sqrt{u}}\) and \(\frac{dg}{dx}=7\). Now, substituting \(u=7x-1\), we get: $$\frac{dy}{dx}=\frac{1}{2\sqrt{7x-1}}\cdot 7$$
05

Simplify the equation

Finally, we can simplify the expression for \(\frac{dy}{dx}\) by multiplying the constants: $$\frac{dy}{dx}=\frac{7}{2\sqrt{7x-1}}$$ The derivative of the function \(y=\sqrt{7x-1}\) with respect to \(x\) is \(\frac{dy}{dx}=\frac{7}{2\sqrt{7x-1}}\).

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Most popular questions from this chapter

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