Step-by-step explanation:
To find the derivative of the given function y = x^x + x^(1/2), we can use the sum rule and the chain rule.
Let's differentiate each term separately:
For the first term, y₁ = x^x:
To differentiate x^x, we can take the natural logarithm of both sides and use implicit differentiation.
ln(y₁) = ln(x^x)
ln(y₁) = x ln(x)
Now, differentiating implicitly with respect to x:
1/y₁ * dy₁/dx = ln(x) + 1
Therefore, dy₁/dx = y₁ * (ln(x) + 1)
For the second term, y₂ = x^(1/2):
Differentiating x^(1/2) with respect to x:
dy₂/dx = (1/2) * x^(-1/2)
dy₂/dx = (1/2) * (1/sqrt(x))
dy₂/dx = 1/(2sqrt(x))
Now, using the sum rule, the derivative of y = y₁ + y₂ is:
dy/dx = dy₁/dx + dy₂/dx
dy/dx = y₁ * (ln(x) + 1) + 1/(2sqrt(x))
Therefore, the derivative of y = x^x + x^(1/2) is dy/dx = y₁ * (ln(x) + 1) + 1/(2sqrt(x)), where y₁ = x^x.
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Verified answer
Step-by-step explanation:
To find the derivative of the given function y = x^x + x^(1/2), we can use the sum rule and the chain rule.
Let's differentiate each term separately:
For the first term, y₁ = x^x:
To differentiate x^x, we can take the natural logarithm of both sides and use implicit differentiation.
ln(y₁) = ln(x^x)
ln(y₁) = x ln(x)
Now, differentiating implicitly with respect to x:
1/y₁ * dy₁/dx = ln(x) + 1
Therefore, dy₁/dx = y₁ * (ln(x) + 1)
For the second term, y₂ = x^(1/2):
Differentiating x^(1/2) with respect to x:
dy₂/dx = (1/2) * x^(-1/2)
dy₂/dx = (1/2) * (1/sqrt(x))
dy₂/dx = 1/(2sqrt(x))
Now, using the sum rule, the derivative of y = y₁ + y₂ is:
dy/dx = dy₁/dx + dy₂/dx
dy/dx = y₁ * (ln(x) + 1) + 1/(2sqrt(x))
Therefore, the derivative of y = x^x + x^(1/2) is dy/dx = y₁ * (ln(x) + 1) + 1/(2sqrt(x)), where y₁ = x^x.