QUESTION 2 Find the derivative of the function
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The Correct Answer and Explanation is:
Correct Answer:
The correct option is the second one:
(6x² + e^x) sinx – (4x³ + 2e^x) cosx / sin³x
Explanation:
To find the derivative of the function f(x) = (2x³ + e^x) / sin²(x), we must use the Quotient Rule. The function is a quotient of two simpler functions. Let’s define them as:
- The numerator: g(x) = 2x³ + e^x
- The denominator: h(x) = sin²(x)
The Quotient Rule states that for a function f(x) = g(x) / h(x), its derivative is:
f'(x) = [g'(x)h(x) – g(x)h'(x)] / [h(x)]²
Step 1: Find the derivative of the numerator, g'(x).
Using the power rule for 2x³ and the rule for the exponential function e^x:
g'(x) = d/dx (2x³ + e^x) = 6x² + e^x
Step 2: Find the derivative of the denominator, h'(x).
The function h(x) = sin²(x) can be written as (sin(x))². We must use the Chain Rule.
Let u = sin(x), so h(x) = u².
The derivative is h'(x) = (d/du)(u²) * (d/dx)(sin(x)).
h'(x) = (2u) * (cos(x))
Substituting u = sin(x) back into the equation:
h'(x) = 2sin(x)cos(x)
Step 3: Apply the Quotient Rule.
Now, we substitute g(x), h(x), g'(x), and h'(x) into the Quotient Rule formula:
f'(x) = [ (6x² + e^x) * sin²(x) – (2x³ + e^x) * (2sin(x)cos(x)) ] / [sin²(x)]²
Step 4: Simplify the expression.
The denominator becomes (sin²(x))² = sin⁴(x).
f'(x) = [ (6x² + e^x)sin²(x) – 2(2x³ + e^x)sin(x)cos(x) ] / sin⁴(x)
We can see a common factor of sin(x) in both terms of the numerator. Let’s factor it out:
f'(x) = [ sin(x) * { (6x² + e^x)sin(x) – 2(2x³ + e^x)cos(x) } ] / sin⁴(x)
Now, we can cancel one sin(x) from the numerator and the denominator:
f'(x) = [ (6x² + e^x)sin(x) – 2(2x³ + e^x)cos(x) ] / sin³(x)
Finally, distribute the 2 in the second term of the numerator:
f'(x) = [ (6x² + e^x)sin(x) – (4x³ + 2e^x)cos(x) ] / sin³(x)
This result matches the second option provided.
