d dx [(e^x + x)^x].
Can someone please show me a step by step of how to do this. Also i'm stuck at (e^x+1)((e^x+x)^x * ln(e^x+x)) ... am i on the right track or?
y = (e^x + x)^x
I would take ln of both sides
ln y = x ln(e^x + x)
now use the product rule
(dy/dx)/x = x(e^x + 1) + ln(e^x + 1)
dy/dx = y(x(e^x + 1) + ln(e^x + 1))
I don't know what kind of simplification you need , but you could sub back the original y value
dy/dx = ( (e^x + x)^x )*(x(e^x + 1) + ln(e^x + 1))
Just checking it on calculator. For some reason the answer is wrong.
To find the derivative of the function f(x) = (e^x + x)^x, we can use the chain rule and the logarithmic differentiation method. Here's a step-by-step approach:
Step 1: Rewrite the function using the exponential form:
f(x) = e^(x * ln(e^x + x))
Step 2: Take the natural logarithm of both sides:
ln(f(x)) = x * ln(e^x + x)
Step 3: Differentiate implicitly with respect to x. For the left side, we use the chain rule:
1/f(x) * f'(x) = ln(e^x + x) + x * (1/(e^x + x)) * (e^x + 1)
Step 4: Solve for f'(x):
f'(x) = f(x) * [ln(e^x + x) + (x * (e^x + 1))/(e^x + x)]
Step 5: Substitute back f(x) into the equation:
f'(x) = (e^(x * ln(e^x + x))) * [ln(e^x + x) + (x * (e^x + 1))/(e^x + x)]
Step 6: Simplify using the properties of logarithms:
f'(x) = (e^(x * ln(e^x + x))) * [ln(e^x + x) + (x * e^x + x)/(e^x + x)]
Step 7: Further simplify by canceling out terms:
f'(x) = (e^(x * ln(e^x + x))) * [ln(e^x + x) + (x * e^x)/(e^x + x) + x/(e^x + x)]
Step 8: Finally, simplify the exponent:
f'(x) = (e^x + x)^[x * ln(e^x + x) + ln(e^x + x) + (x * e^x)/(e^x + x) + x/(e^x + x)]
You can stop here or further simplify, depending on your requirements.
Regarding your attempt:
(e^x+1)((e^x+x)^x * ln(e^x+x))
Your evaluation is partially correct, but it seems you missed some simplifications along the way, and the final expression is not completely accurate. It's important to simplify the terms and apply the properties of logarithms correctly.
I hope this step-by-step explanation helps you understand the process of finding the derivative of (e^x + x)^x.