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Answer :
Final Answer:
The correct answer is **c. (eˣ(5x⁴-3x³)+xeˣ(20x³-9x²))/(5x⁴-3x³)²**
Explanation:
To find the derivative of the given function f(x) = (xeˣ)/(5x⁴-3x³), we can use the quotient rule. The quotient rule states that if you have a function g(x) = h(x)/j(x), then g'(x) = (h'(x)j(x) - h(x)j'(x))/[j(x)]².
Let's apply this rule to f(x) = (xeˣ)/(5x⁴-3x³):
1. Find h'(x): Derivative of xeˣ with respect to x is eˣ + xeˣ.
2. Find j'(x): Derivative of (5x⁴-3x³) with respect to x is (20x³ - 9x²).
Now, apply the quotient rule:
f'(x) = [(eˣ + xeˣ)(5x⁴-3x³) - (xeˣ)(20x³ - 9x²)] / [(5x⁴-3x³)²]
Simplify the expression:
f'(x) = (eˣ(5x⁴-3x³) + xeˣ(20x³ - 9x²)) / (5x⁴-3x³)²
Therefore, the correct answer is option c. (eˣ(5x⁴-3x³) + xeˣ(20x³ - 9x²)) / (5x⁴-3x³)².
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Final answer:
To find the derivative of the function f(x) = (xeˣ)/(5x⁴-3x³), we can use the quotient rule. Applying the quotient rule correctly yields the answer a. (eˣ(5x⁴-3x³) + xeˣ(20x³-9x²))/(5x⁴-3x³)².
Explanation:
To find the derivative of the function f(x) = (xeˣ)/(5x⁴-3x³), we will use the quotient rule.
The quotient rule states that for two functions u(x) and v(x), the derivative of their quotient is given by (u'(x)v(x) - u(x)v'(x))/(v(x))².
Let's apply this rule to our function:
Let u(x) = xeˣ and v(x) = 5x⁴-3x³.
The derivatives of u(x) and v(x) are u'(x) = eˣ + xeˣ and v'(x) = 20x³-9x², respectively.
Using the quotient rule, we have f'(x) = (u'(x)v(x) - u(x)v'(x))/(v(x))² = (eˣ(5x⁴-3x³) + xeˣ(20x³-9x²))/(5x⁴-3x³)².
Therefore, the correct answer is a. (eˣ(5x⁴-3x³) + xeˣ(20x³-9x²))/(5x⁴-3x³)².
