homework5: problem 15 (1 point) find an equation of the line tangent to the graph of y = \\frac{6}{\\sqrt{x}}…
homework5: problem 15 (1 point) find an equation of the line tangent to the graph of y = \\frac{6}{\\sqrt{x}} at the point (1, 6/1). answer: y =
Answer
Explanation:
Step1: Rewrite the function
Rewrite $y = \frac{6}{\sqrt{x}}$ as $y=6x^{-\frac{1}{2}}$.
Step2: Find the derivative
Using the power - rule $(x^n)'=nx^{n - 1}$, we have $y'=6\times(-\frac{1}{2})x^{-\frac{1}{2}-1}=-3x^{-\frac{3}{2}}=-\frac{3}{x^{\frac{3}{2}}}$.
Step3: Find the slope of the tangent line at the given point
Substitute $x = 1$ into the derivative. When $x = 1$, $y'(1)=-\frac{3}{1^{\frac{3}{2}}}=-3$.
Step4: Use the point - slope form of a line
The point - slope form is $y - y_1=m(x - x_1)$, where $(x_1,y_1)=(1,6)$ and $m=-3$. So $y - 6=-3(x - 1)$.
Step5: Simplify the equation
Expand the right - hand side: $y-6=-3x + 3$. Then $y=-3x+9$.
Answer:
$y=-3x + 9$