shamieh said:Find the equation of the tangent line to g(x) = arctan(x) + ln(x) @ x = 1
\(\displaystyle y - \frac{\pi}{4} = \frac{3}{2}(x - 1)\)
Deveno said:Your answer is indeed correct, but as no actual work is shown...
The equation of a tangent line is a mathematical representation of a line that touches a curve at one point. It can be written in the form of y = mx + b, where m is the slope of the line and b is the y-intercept.
To find the equation of a tangent line, you first need to find the slope of the curve at the point of tangency. This can be done by taking the derivative of the curve at that point. Then, you can use the point-slope form of a line (y - y1 = m(x - x1)) to find the equation of the tangent line, where (x1, y1) is the point of tangency and m is the slope.
The point-slope form of a line is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope of the line. This form is useful for finding the equation of a line when you know the slope and a point on the line.
The derivative of a curve can be found by using the rules of differentiation, depending on the type of function. For example, to find the derivative of a polynomial function, you can use the power rule. To find the derivative of a trigonometric function, you can use the chain rule. It is important to understand these rules and practice using them to take derivatives.
The tangent line to a curve is significant because it represents the instantaneous rate of change of the curve at a specific point. This is useful in many applications, such as finding the velocity of an object at a specific time or the growth rate of a population at a specific point. It also helps us understand the behavior of a curve and make predictions about its future behavior.