a. Find an equation of the tangent line at $x = a$ .
b. Use a graphing utility to graph the curve and the tangent line on the same set of axes.
$y = e^{x}; a = \ln 10$
a. At $x = a$ , an equation of the tangent line is $y =$

Answer

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Answer

$y = 10 x - 13.02585092994046$ is the equation of the tangent line at $x = a$ .

Steps

Step 1 ：We are given the function $y = e^{x}$ and $a = \ln 10$ .

Step 2 ：The equation of the tangent line to the function $f (x)$ at $x = a$ is given by $y = f (a) + f^{'} (a) (x - a)$ .

Step 3 ：First, we need to find the value of $f (a)$ and $f^{'} (a)$ .

Step 4 ：Since $f (x) = e^{x}$ , we have $f (a) = e^{\ln 10} = 10$ .

Step 5 ：The derivative of $f (x) = e^{x}$ is also $e^{x}$ , so $f^{'} (a) = e^{\ln 10} = 10$ .

Step 6 ：Substituting these values into the equation of the tangent line, we get $y = 10 + 10 (x - \ln 10)$ .

Step 7 ：Simplifying this equation, we get $y = 10 x - 13.02585092994046$ .

Step 8 ： $y = 10 x - 13.02585092994046$ is the equation of the tangent line at $x = a$ .