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If y=sin((1+x2)/(1-x2)), (dy/dx)=
- a)[(4x)/(1-x2)]. cos[(1+x2)/(1-x2)]
- b)[(x)/(1-x2)2]. cos[(1+x2)/(1-x2)]
- c)[(x)/(1-x2)]. cos[(1+x2)/(1-x2)]
- d)4x × cos((1 + x²) / (1 − x²)) / (1 − x²)²
Correct answer is option 'D'. Can you explain this answer?
Most Upvoted Answer
If y=sin((1+x2)/(1-x2)), (dy/dx)=a)[(4x)/(1-x2)]. cos[(1+x2)/(1-x2)]b)...
Given:
y = sin((1 + x²) / (1 − x²))
y = sin((1 + x²) / (1 − x²))
We are to find dy/dx.
Step 1: Identify inner and outer functions
Let:
u = (1 + x²) / (1 − x²) ← inner function
y = sin(u) ← outer function
u = (1 + x²) / (1 − x²) ← inner function
y = sin(u) ← outer function
Step 2: Differentiate using chain rule
Using the chain rule:
dy/dx = dy/du × du/dx
dy/dx = dy/du × du/dx
We know:
dy/du = cos(u)
dy/du = cos(u)
Step 3: Differentiate u
To find du/dx for u = (1 + x²) / (1 − x²), use the quotient rule:
du/dx = [(1 − x²)(d/dx(1 + x²)) − (1 + x²)(d/dx(1 − x²))] / (1 − x²)²
Now compute derivatives:
d/dx(1 + x²) = 2x
d/dx(1 − x²) = −2x
d/dx(1 + x²) = 2x
d/dx(1 − x²) = −2x
So:
du/dx = [(1 − x²)(2x) − (1 + x²)(−2x)] / (1 − x²)²
du/dx = [(1 − x²)(2x) − (1 + x²)(−2x)] / (1 − x²)²
Simplify numerator:
= (1 − x²)(2x) + (1 + x²)(2x)
= 2x(1 − x² + 1 + x²)
= 2x(2)
= 4x
= (1 − x²)(2x) + (1 + x²)(2x)
= 2x(1 − x² + 1 + x²)
= 2x(2)
= 4x
So:
du/dx = 4x / (1 − x²)²
du/dx = 4x / (1 − x²)²
Step 4: Combine the results
dy/dx = cos((1 + x²) / (1 − x²)) × 4x / (1 − x²)²
Final Answer:
dy/dx = 4x × cos((1 + x²) / (1 − x²)) / (1 − x²)²
dy/dx = 4x × cos((1 + x²) / (1 − x²)) / (1 − x²)²
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If y=sin((1+x2)/(1-x2)), (dy/dx)=a)[(4x)/(1-x2)]. cos[(1+x2)/(1-x2)]b)...
Understanding the Function
We start with the function:
y = sin((1 + x²) / (1 - x²))
To find dy/dx, we will use the chain rule and quotient rule of differentiation.
Applying the Chain Rule
1. The outer function is sin(u), where u = (1 + x²) / (1 - x²).
2. The derivative of sin(u) is cos(u) * du/dx.
Finding du/dx
Now we need to differentiate u:
- u = (1 + x²) / (1 - x²)
Using the quotient rule:
- If f = 1 + x² and g = 1 - x², then:
du/dx = (g * f' - f * g') / g²
Where:
- f' = 2x (derivative of f)
- g' = -2x (derivative of g)
Substituting:
du/dx = [(1 - x²)(2x) - (1 + x²)(-2x)] / (1 - x²)²
This simplifies to:
du/dx = [2x(1 - x² + 1 + x²)] / (1 - x²)²
Thus:
du/dx = [4x] / (1 - x²)²
Final Derivative Calculation
Now substituting back into dy/dx:
dy/dx = cos((1 + x²) / (1 - x²)) * du/dx
Therefore, we have:
dy/dx = cos((1 + x²) / (1 - x²)) * [4x / (1 - x²)²]
Conclusion
Thus, the final derivative is:
dy/dx = (4x * cos((1 + x²) / (1 - x²))) / (1 - x²)²
This matches option 'D', confirming its correctness.
We start with the function:
y = sin((1 + x²) / (1 - x²))
To find dy/dx, we will use the chain rule and quotient rule of differentiation.
Applying the Chain Rule
1. The outer function is sin(u), where u = (1 + x²) / (1 - x²).
2. The derivative of sin(u) is cos(u) * du/dx.
Finding du/dx
Now we need to differentiate u:
- u = (1 + x²) / (1 - x²)
Using the quotient rule:
- If f = 1 + x² and g = 1 - x², then:
du/dx = (g * f' - f * g') / g²
Where:
- f' = 2x (derivative of f)
- g' = -2x (derivative of g)
Substituting:
du/dx = [(1 - x²)(2x) - (1 + x²)(-2x)] / (1 - x²)²
This simplifies to:
du/dx = [2x(1 - x² + 1 + x²)] / (1 - x²)²
Thus:
du/dx = [4x] / (1 - x²)²
Final Derivative Calculation
Now substituting back into dy/dx:
dy/dx = cos((1 + x²) / (1 - x²)) * du/dx
Therefore, we have:
dy/dx = cos((1 + x²) / (1 - x²)) * [4x / (1 - x²)²]
Conclusion
Thus, the final derivative is:
dy/dx = (4x * cos((1 + x²) / (1 - x²))) / (1 - x²)²
This matches option 'D', confirming its correctness.
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Question Description
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