Questions: Find an equation of the tangent line to the curve at the given point. y=5x-4√x,(1,1) y=

Solution Steps

To find the equation of the tangent line, we first need to find the derivative of the function \( y = 5x - 4\sqrt{x} \). The derivative, \( y' \), will give us the slope of the tangent line at any point \( x \).

The derivative of \( 5x \) is \( 5 \).

The derivative of \( -4\sqrt{x} \) is found using the power rule. Rewrite \( \sqrt{x} \) as \( x^{1/2} \), so the derivative is:

\[ \frac{d}{dx}(-4x^{1/2}) = -4 \cdot \frac{1}{2}x^{-1/2} = -2x^{-1/2} = -\frac{2}{\sqrt{x}} \]

Thus, the derivative of the function is:

\[ y' = 5 - \frac{2}{\sqrt{x}} \]

We need to find the slope of the tangent line at the point \( (1, 1) \). Substitute \( x = 1 \) into the derivative:

\[ y'(1) = 5 - \frac{2}{\sqrt{1}} = 5 - 2 = 3 \]

The slope of the tangent line at \( (1, 1) \) is \( 3 \).

The point-slope form of a line is given by:

\[ y - y_1 = m(x - x_1) \]

where \( m \) is the slope and \( (x_1, y_1) \) is the point on the line. Here, \( m = 3 \) and \( (x_1, y_1) = (1, 1) \).

Substitute these values into the point-slope form:

\[ y - 1 = 3(x - 1) \]

Simplify the equation:

\[ y - 1 = 3x - 3 \]

\[ y = 3x - 2 \]

Final Answer