Given:
y = log(x)

To find:
[dy/dx]x=1

Solution:

To find the derivative of y with respect to x, we can use the logarithmic differentiation method.

Step 1: Rewrite the equation:
y = log(x)
This can be rewritten as:
x = 10^y

Step 2: Take the natural logarithm of both sides:
ln(x) = ln(10^y)

Step 3: Apply the logarithmic property:
ln(x) = y * ln(10)

Step 4: Differentiate both sides with respect to x:
(1/x) * dx/dy = ln(10)

Step 5: Solve for dy/dx:
dx/dy = (x * ln(10))

Since we want to find [dy/dx]x=1, we need to substitute x = 1 into the above equation.

dy/dx = (1 * ln(10))
dy/dx = ln(10)

Therefore, [dy/dx]x=1 = ln(10).

Explanation:
The given equation is y = log(x), where y represents the logarithm of x. We need to find the derivative of y with respect to x at x = 1.

To find the derivative, we use the logarithmic differentiation method. By rewriting the equation as x = 10^y, we can take the natural logarithm of both sides to simplify the equation. Then, by differentiating both sides with respect to x and solving for dy/dx, we find that dy/dx = ln(10).

Substituting x = 1 into the equation, we get [dy/dx]x=1 = ln(10). Therefore, the derivative of y with respect to x at x = 1 is ln(10).

Summary:
The derivative of y = log(x) with respect to x at x = 1 is ln(10).