logarithmic differentiation of find derivative of y= square root of x^x
To find the derivative of the function y = sqrt(x^x), we can use logarithmic differentiation. Here's how you can do it step by step:
Step 1: Take the natural logarithm (ln) of both sides of the equation. This step allows us to convert the power of x into a product, which simplifies the differentiation process. Recall the logarithmic property that ln(a^b) = b * ln(a).
ln(y) = ln(sqrt(x^x))
Step 2: Apply the logarithmic properties to simplify the expression. By using the square root property, we can rewrite the equation as:
ln(y) = ln(x^(x/2))
Step 3: Apply the power rule of logarithms, which states that ln(a^b) = b * ln(a). Differentiate both sides of the equation with respect to x:
(d/dx) ln(y) = (d/dx) [ln(x^(x/2))]
Step 4: Use the chain rule on the right side. When we differentiate ln of a function, we multiply it by the derivative of the function. So the derivative of ln(x) is 1/x. Applying the chain rule, we have:
(d/dx) ln(y) = (d/dx) [(x/2) * ln(x)]
= (d/dx) (x/2) * ln(x) + (x/2) * (d/dx) ln(x)
= (1/2) * ln(x) + (x/2) * (1/x)
Step 5: Simplify the expression obtained in the previous step:
(d/dx) ln(y) = (1/2) * ln(x) + 1/2
Step 6: Now we have the derivative of ln(y) with respect to x. To find the derivative of y, we need to multiply this by dy/dx:
dy/dx * (1/y) = (1/2) * ln(x) + 1/2
Step 7: Rearrange the equation to solve for dy/dx, which is our desired derivative:
dy/dx = y * [(1/2) * ln(x) + 1/2]
Step 8: Substitute the value of y back into the equation. Since y = sqrt(x^x), we have:
dy/dx = sqrt(x^x) * [(1/2) * ln(x) + 1/2]
Therefore, the derivative of y = sqrt(x^x) is:
dy/dx = sqrt(x^x) * [(1/2) * ln(x) + 1/2]