Question
Find the derinative 1) \( y=\frac{e^{x}}{5 x-3} \)
Ask by Parsons Maxwell. in the United States
Mar 11,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The derivative of \( y = \frac{e^{x}}{5x-3} \) is \( y' = \frac{e^{x}(5x-8)}{(5x-3)^2} \).
Solution
Let \( y = \frac{e^x}{5x-3} \).
**Step 1.** Identify the numerator and denominator of the function:
\[
u(x) = e^x, \quad v(x) = 5x-3.
\]
**Step 2.** Compute the derivatives of \( u(x) \) and \( v(x) \):
\[
u'(x) = e^x, \quad v'(x) = 5.
\]
**Step 3.** Apply the quotient rule which states:
\[
y' = \frac{u'v - uv'}{v^2}.
\]
Substitute the functions and their derivatives:
\[
y' = \frac{e^x(5x-3) - e^x(5)}{(5x-3)^2}.
\]
**Step 4.** Factor out the common term \( e^x \) in the numerator:
\[
y' = \frac{e^x\left[(5x-3)-5\right]}{(5x-3)^2}.
\]
**Step 5.** Simplify the expression inside the brackets:
\[
(5x-3)-5 = 5x-8.
\]
Thus, the derivative is:
\[
y' = \frac{e^x(5x-8)}{(5x-3)^2}.
\]
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Simplify this solution Beyond the Answer
To find the derivative of \( y = \frac{e^{x}}{5x - 3} \), we will use the quotient rule, which states that if you have a function \( y = \frac{u}{v} \), then the derivative \( y' \) is given by \( y' = \frac{u'v - uv'}{v^2} \). Here, \( u = e^{x} \) and \( v = 5x - 3 \). First, we find the derivatives: - \( u' = e^{x} \) - \( v' = 5 \) Now, applying the quotient rule: \[ y' = \frac{(e^{x})(5x - 3) - (e^{x})(5)}{(5x - 3)^2} \] This simplifies to: \[ y' = \frac{e^{x}(5x - 3 - 5)}{(5x - 3)^2} = \frac{e^{x}(5x - 8)}{(5x - 3)^2} \] So the derivative is: \[ y' = \frac{e^{x}(5x - 8)}{(5x - 3)^2} \]
