Find the Tangent Line at the Point square root of x , (1,1)

Problem

ƒ(x)=√(,x),(1,1)

Solution

1. Identify the function and the point of tangency ((x_0),(y_0))=(1,1)

2. Rewrite the function in power form to prepare for differentiation.

ƒ(x)=x(1/2)

3. Apply the power rule to find the derivative, which represents the slope of the tangent line at any point x

d(√(,x))/d(x)=1/2*x(−1/2)

4. Simplify the derivative expression.

ƒ(x)′=1/(2√(,x))

5. Evaluate the derivative at x=1 to find the slope m

m=ƒ(1)′=1/(2√(,1))

m=1/2

6. Use the point-slope form y−(y_0)=m*(x−(x_0)) to write the equation of the line.

y−1=1/2*(x−1)

7. Solve for y to put the equation in slope-intercept form.

y−1=1/2*x−1/2

y=1/2*x+1/2

Final Answer

y=1/2*x+1/2

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