3 Differentiate the expression (x + y )" with respect to x. (Use D for 2). dx
Advanced Engineering Mathematics
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Rate of Change
The relation between two quantities which displays how much greater one quantity is than another is called ratio.
Slope
The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:
Question
![**Differentiation Problem:**
Differentiate the expression \((x^2 + y^2)^3\) with respect to \(x\). (Use \(D\) for \(\frac{dy}{dx}\)).
**The result is:** [ ]
**Explanation:**
To differentiate the expression \((x^2 + y^2)^3\) with respect to \(x\), you will need to use the chain rule. The chain rule in calculus is a method for finding the derivative of a composite function. In this case, the outer function is the cube, and the inner function is \(x^2 + y^2\).
1. Let \(u = x^2 + y^2\).
2. The expression becomes \(u^3\).
3. Use the chain rule:
- Differentiate the outer function: \(\frac{d}{du}(u^3) = 3u^2\).
- Differentiate the inner function with respect to \(x\): \(\frac{d}{dx}(x^2 + y^2) = 2x + 2y \frac{dy}{dx}\).
4. Combine these results:
- \(\frac{d}{dx}((x^2 + y^2)^3) = 3(x^2 + y^2)^2 \cdot (2x + 2y \frac{dy}{dx})\).
Insert this result into the solution box.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F27415415-90e2-442e-a375-49c282576b25%2Ffc392f15-5518-4bee-ac2b-d187dd15ebff%2Flz0z8zs_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Differentiation Problem:**
Differentiate the expression \((x^2 + y^2)^3\) with respect to \(x\). (Use \(D\) for \(\frac{dy}{dx}\)).
**The result is:** [ ]
**Explanation:**
To differentiate the expression \((x^2 + y^2)^3\) with respect to \(x\), you will need to use the chain rule. The chain rule in calculus is a method for finding the derivative of a composite function. In this case, the outer function is the cube, and the inner function is \(x^2 + y^2\).
1. Let \(u = x^2 + y^2\).
2. The expression becomes \(u^3\).
3. Use the chain rule:
- Differentiate the outer function: \(\frac{d}{du}(u^3) = 3u^2\).
- Differentiate the inner function with respect to \(x\): \(\frac{d}{dx}(x^2 + y^2) = 2x + 2y \frac{dy}{dx}\).
4. Combine these results:
- \(\frac{d}{dx}((x^2 + y^2)^3) = 3(x^2 + y^2)^2 \cdot (2x + 2y \frac{dy}{dx})\).
Insert this result into the solution box.
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