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Chapter 3: Problem 63

Use the Chain Rule combined with other differentiation rules to find the derivative of the following functions. $$y=\sqrt{x^{4}+\cos 2 x}$$

Short Answer

Expert verified
Question: Find the derivative of the function \(y = \sqrt{x^4 + \cos(2x)}\) with respect to x. Answer: \(\frac{dy}{dx} = \dfrac{1}{2}(x^4 + \cos(2x))^{-\frac{1}{2}}(4x^3 - 2\sin(2x))\)

Step by step solution

01

Identify the Outer Function and Inner Function

Based on our given function, we can identify the outer function and inner function as follows: Outer function: \(f(u) = \sqrt{u}\) Inner function: \(u(x) = x^4 + \cos2x\)
02

Apply the Chain Rule

To apply the Chain Rule, we can follow the formula: $$\frac{dy}{dx} = \frac{df(u)}{du} \cdot \frac{du}{dx}$$ Next, we need to find the derivatives of outer (f(u)) and inner (u(x)) functions.
03

Differentiate The Outer function

The outer function is a square root function, which can also be written as: $$f(u) = u^{\frac{1}{2}}$$ Differentiating with respect to u, we obtain: $$\frac{df(u)}{du} = \frac{1}{2}u^{-\frac{1}{2}}$$
04

Differentiate The Inner Function

The inner function, u(x), consists of two terms: \(x^4\) and \(\cos(2x)\). We will differentiate each term separately. Term 1: \(x^4\) $$\frac{d(x^4)}{dx} = 4x^3$$ Term 2: \(\cos(2x)\) Using the Chain Rule, we have: $$\frac{d(\cos(2x))}{dx} = -\sin(2x) \cdot 2$$ $$\frac{d(\cos(2x))}{dx} = -2\sin(2x)$$ Now, adding the derivatives of both terms, we have: $$\frac{du}{dx} = 4x^3 - 2\sin(2x)$$
05

Find The Derivative of the Given Function

Now that we have found the derivatives of the outer function and the inner function, we can apply the Chain Rule: $$\frac{dy}{dx} = \frac{df(u)}{du} \cdot \frac{du}{dx} = \frac{1}{2}u^{-\frac{1}{2}} \cdot (4x^3 - 2\sin(2x))$$ Substitute u(x) back into the derivative: $$\frac{dy}{dx} = \frac{1}{2}(x^4 + \cos(2x))^{-\frac{1}{2}} \cdot (4x^3 - 2\sin(2x))$$ This is the derivative of the given function.

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