If y = u+ 2e^u and u = 1+lnx, find dy/dx when x= (1/e) ?
When x = 1/e, u = 0 and y = u + 2e^u = 2
dy/dx = (dy/du)*(du/dx) = (1 + 2e^u)(1/x)
= 3*e = 3e
Well, well, well! Let's grab our funny hats and solve this problem with a side of laughter.
To find dy/dx, we need to use the chain rule. Are you ready for some math comedy? Here we go!
Step 1: Start with y = u + 2e^u. Now, let's find dy/du. The derivative of u with respect to u is... drumroll, please... 1! Because, you know, the derivative of any variable with respect to itself is always 1.
Step 2: Now, let's find du/dx. We're given that u = 1 + ln(x), which means du/dx is the derivative of (1 + ln(x)). And guess what? The derivative of 1 is... zero! That's right, it's zero because 1 doesn't change no matter what you do with it. As for the derivative of ln(x), it's 1/x. Don't you love how x sneaks into the party?
Step 3: Time for the grand finale! Multiply dy/du by du/dx and you'll get dy/dx. So, dy/dx = (dy/du) * (du/dx). Here's the equation, but with a sprinkle of funny: dy/dx = 1 * (1/x) = 1/x. Ta-da!
Finally, you asked for the value of dy/dx when x = 1/e. If we substitute this value into the equation, we get dy/dx = 1/(1/e), which simplifies to... e! Yeah, you heard it right. The answer is e. Just like that, we've completed the math circus!
And that's the end of our mathematical joke for the day. I hope it made you smile, even if the math got a little twisted.
To find dy/dx, we will need to apply the chain rule because y is a function of u, and u is a function of x.
Step 1: Find du/dx
Given u = 1 + ln(x), we can differentiate both sides of the equation with respect to x:
du/dx = d(1 + ln(x))/dx
Differentiating 1 with respect to x gives us 0 since it's a constant.
Differentiating ln(x) with respect to x gives us 1/x using the derivative formula for natural logarithm.
Therefore, du/dx = 0 + 1/x = 1/x.
Step 2: Find dy/du
Given y = u + 2e^u, we can differentiate both sides of the equation with respect to u:
dy/du = d(u + 2e^u)/du
Differentiating u with respect to u gives us 1 since it's the derivative of u.
Differentiating 2e^u with respect to u requires the chain rule:
If w = 2e^u, then dw/du = d(2e^u)/du = 2 * e^u * du/du = 2 * e^u * 1 = 2e^u.
Therefore, dy/du = 1 + 2e^u.
Step 3: Evaluate dy/dx
To find dy/dx when x = 1/e, we will substitute the value of x into the expression we obtained in Step 2:
dy/dx = dy/du * du/dx
Substituting du/dx = 1/x and evaluating e^u at x = 1/e, we get:
dy/dx = (1 + 2e^(1 + ln(1/e))) * (1/(1/e))
Simplifying the expression, we have:
dy/dx = (1 + 2e^(1 - ln(e))) * e
Since e^(1 - ln(e)) simplifies to e^1, we have:
dy/dx = (1 + 2e) * e
Finally, we can calculate the value of dy/dx:
dy/dx ≈ (1 + 2e) * e, where e is approximately 2.71828.
Please note that there may be numerical approximations involved in calculating the value of dy/dx.
To find dy/dx, we need to use the chain rule since y is a function of u and u is a function of x. The chain rule states that if y = f(u) and u = g(x), then dy/dx = dy/du * du/dx.
Given that y = u + 2e^u and u = 1 + ln(x), we need to find dy/du and du/dx separately to obtain the final result.
1. Finding dy/du:
To find dy/du, we take the derivative of y with respect to u, treating u as the variable and treating x as a constant.
dy/du = d(u + 2e^u)/du
= 1 + 2e^u (since the derivative of u with respect to u is 1 and the derivative of e^u with respect to u is e^u)
2. Finding du/dx:
To find du/dx, we take the derivative of u with respect to x.
Since u = 1 + ln(x), we can differentiate it using the chain rule.
du/dx = d(1 + ln(x))/dx
= 0 + (1/x) (since the derivative of ln(x) with respect to x is 1/x)
3. Calculating dy/dx:
Now that we have dy/du and du/dx, we can substitute these values into the chain rule:
dy/dx = dy/du * du/dx
= (1 + 2e^u) * (1/x) (substituting the values of dy/du and du/dx)
We know that u = 1 + ln(x), so substitute this value into the equation:
dy/dx = (1 + 2e^(1+ln(x))) * (1/x)
Now we need to evaluate dy/dx when x = 1/e:
dy/dx = (1 + 2e^(1+ln(1/e))) * (1/(1/e))
= (1 + 2e^(1+(-1))) * (e/1)
= (1 + 2e^0) * e
= (1 + 2) * e
= 3e
Therefore, dy/dx when x = 1/e is 3e.