Question

find an equation of the tangent line to the curve ( y = xsqrt{x} ) that is parallel to the line ( y = 9 + 9x ).( y = )

Explanation:

Step1: Simplify the curve equation

First, rewrite \( y = x\sqrt{x} \) as \( y = x^{3/2} \) since \( \sqrt{x}=x^{1/2} \) and \( x\times x^{1/2}=x^{1 + 1/2}=x^{3/2} \).

Step2: Find the derivative of the curve

Using the power rule for differentiation, if \( y = x^{n} \), then \( y'=nx^{n - 1} \). For \( y = x^{3/2} \), the derivative \( y'=\frac{3}{2}x^{\frac{3}{2}-1}=\frac{3}{2}x^{1/2}=\frac{3}{2}\sqrt{x} \).

Step3: Determine the slope of the tangent line

The line \( y = 9 + 9x \) has a slope of \( 9 \) (since it is in the form \( y=mx + b \), where \( m \) is the slope). Since the tangent line is parallel to this line, their slopes are equal. So we set the derivative equal to \( 9 \):
\[
\frac{3}{2}\sqrt{x}=9
\]
Solve for \( x \):
Multiply both sides by \( \frac{2}{3} \): \( \sqrt{x}=9\times\frac{2}{3}=6 \)
Then square both sides: \( x = 6^{2}=36 \)

Step4: Find the corresponding y - value on the curve

Substitute \( x = 36 \) into the original curve equation \( y = x\sqrt{x} \). First, \( \sqrt{36}=6 \), so \( y=36\times6 = 216 \).

Step5: Use the point - slope form to find the tangent line equation

The point - slope form of a line is \( y - y_{1}=m(x - x_{1}) \), where \( (x_{1},y_{1})=(36,216) \) and \( m = 9 \).
Substitute the values: \( y - 216=9(x - 36) \)
Expand the right - hand side: \( y - 216=9x-324 \)
Add 216 to both sides: \( y=9x-324 + 216 \)
Simplify: \( y=9x - 108 \)

Answer:

\( y = 9x-108 \)