Question

\(\left(\frac{x}{a}\right)^{n}+\left(\frac{y}{b}\right)^{n}=2\) $\frac{n}{a}\left(\frac{x}{a}\right)^{n-1}+\frac{n}{b}\left(\frac{y}{b}\right)^{n-t} \frac{d y}{d x}=0$ \(\frac{d y}{d x}=-\frac{b^{n} x^{n-1}}{a^{n} y^{n-1}}=-\frac{b}{a}\)

Short answer

Question: Confirm that the given derivative is accurate for the function \(\left(\frac{x}{a}\right)^{n}+\left(\frac{y}{b}\right)^{n}=2\). Solution: We found that the derivative of the given function with respect to x is \(\frac{dy}{dx} = -\frac{b^{n} x^{n-1}}{a^{n} y^{n-1}}\), which matches the given expression \(\frac{dy}{dx} = -\frac{b}{a}\). Therefore, we can confirm that the given derivative is accurate for the function.

Step-by-step solution

Differentiate the given function w.r.t x

To find the derivative, we'll apply the chain rule. So, we will first differentiate the left side of the equation with respect to x, and then do the same for the right side: \( \frac{d}{dx} \left[\left(\frac{x}{a}\right)^{n}+\left(\frac{y}{b}\right)^{n}\right] = \frac{d}{dx}(2)\) Now, we already know that the derivative of a constant is 0. Next, we will find the derivative of the sum of two terms.

Differentiate the first term

We will differentiate the first term \((\frac{x}{a})^{n}\) with respect to x using the power rule: \(\frac{d}{dx} \left(\frac{x}{a}\right)^{n}= n(\frac{x}{a})^{n-1} \frac{d}{dx} (\frac{x}{a})\) \(= n(\frac{x}{a})^{n-1} \frac{1}{a}\) \(= \frac{n}{a} (\frac{x}{a})^{n-1}\) Now we have the derivative of the first term.

Differentiate the second term

Now we will differentiate the second term \(\left(\frac{y}{b}\right)^{n}\) with respect to x using the chain rule: \(\frac{d}{dx}\left(\frac{y}{b}\right)^{n}=n\left(\frac{y}{b}\right)^{n-1}\frac{d}{dx}\left(\frac{y}{b}\right)\) \(= n \left(\frac{y}{b}\right)^{n-1} \frac{d y}{d x} \frac{1}{b}\) \(= \frac{n}{b} \left(\frac{y}{b}\right)^{n-1} \frac{dy}{dx}\) Now we have the derivative of the second term.

Combining the derivatives

Now we can combine the derivatives from steps 2 and 3: \(\frac{n}{a} (\frac{x}{a})^{n-1} + \frac{n}{b} \left(\frac{y}{b}\right)^{n-1} \frac{dy}{dx} = 0\)

Isolate \(\frac{dy}{dx}\) and simplify

Now we will isolate \(\frac{dy}{dx}\) : \(\frac{n}{b} \left(\frac{y}{b}\right)^{n-1} \frac{dy}{dx} = -\frac{n}{a} (\frac{x}{a})^{n-1}\) \(\frac{dy}{dx} = -\frac{b \cdot n (\frac{x}{a})^{n-1}}{n (\frac{y}{b})^{n-1}}\) Now we will cancel n from both the numerator and the denominator: \(\frac{dy}{dx} = -\frac{b (\frac{x}{a})^{n-1}}{ (\frac{y}{b})^{n-1}}\) By manipulating the exponents, we get: \(\frac{dy}{dx} = -\frac{b^{n} x^{n-1}}{a^{n} y^{n-1}}\)

Compare our derivative to the given one

Now that we have found the derivative of the given function, we can compare it to the given expression: Our derivative: \(\frac{dy}{dx} = -\frac{b^{n} x^{n-1}}{a^{n} y^{n-1}}\) Given derivative: \(\frac{dy}{dx} = -\frac{b^{n} x^{n-1}}{a^{n} y^{n-1}} = -\frac{b}{a}\) Since our derivative matches the given one, this confirms that the derivative is accurate.