Introduction to Tangent Lines and Derivatives

Tangent lines are crucial in calculus for understanding the local behavior of functions at specific points. This article will delve into finding the equation of the tangent line to the curve ( y sqrt{x} ) at the point where ( x 4 ). We’ll explore the derivative concept and its application in determining the slope of the tangent line, followed by the steps to derive the equation of the tangent line.

Understanding the Derivative of y √x

The derivative of a function at a specific point is the slope of the tangent line at that point. For the function ( y sqrt{x} ), we can write the function in index form as ( y x^{1/2} ).

In calculus, the derivative of ( x^n ) is given by ( nx^{n-1} ). Therefore, the derivative of ( y x^{1/2} ) is:

Derivation

Step 1: Express the function in index form

[ y x^{1/2} ]

Step 2: Apply the power rule

[ frac{dy}{dx} frac{1}{2} x^{-1/2} ]

Step 3: Simplify the expression

[ frac{dy}{dx} frac{1}{2sqrt{x}} ]

Evaluating the Derivative at x 4

To find the slope of the tangent line at ( x 4 ), we substitute ( x 4 ) into the derivative:

[ frac{dy}{dx} bigg|_{x4} frac{1}{2sqrt{4}} frac{1}{4} ]

Equation of the Tangent Line

The equation of the tangent line in point-slope form is given by:

[ y - y_0 m(x - x_0) ]

where ( m ) is the slope of the tangent line, and ( (x_0, y_0) ) is the point of tangency.

Step 1: Identify the point of tangency

[ x_0 4, quad y_0 sqrt{4} 2 ]

Step 2: Substitute the slope and the point of tangency into the equation

[ y - 2 frac{1}{4}(x - 4) ]

Step 3: Simplify to get the equation of the tangent line in standard form

[ y - 2 frac{1}{4}x - 1 ]

[ y frac{1}{4}x - 1 2 ]

[ y frac{1}{4}x 1 ]

General Approach for Tangent Line Equations

The process we used to find the equation of the tangent line at a specific point can be generalized:

1. **Find the derivative of the function.**

2. **Evaluate the derivative at the given point to find the slope of the tangent line.**

3. **Identify the point of tangency.**

4. **Use the point-slope form to write the equation of the tangent line.**

5. **Simplify to get the equation in standard form if needed.**

Conclusion

The equation of the tangent line to the curve ( y sqrt{x} ) at ( x 4 ) is:

[ y frac{1}{4}x 1 ]

This article has provided a detailed step-by-step guide to finding the tangent line equation using the derivatives and the point-slope form. Understanding these concepts is fundamental in exploring applications of calculus in various fields.