Find the slope m of the tangent line to the graph of the function at the given point and determine an equation of the tangent line.
f(x) = 15/4 x at (1,15/4)
The lack of parentheses makes it impossible for me to say if x is in the numerator or the denominator.
If x is in the numerator y =(15 x/4) the problem is trivial The slope is 15/4 for all x.
If you really mean y = 15 /(4x) than
dy/dx = -15 /(16x^2)
which for x = 1 is
slope = -15/16
To find the slope of the tangent line to the graph of the function at a given point, we need to find the derivative of the function and evaluate it at that point. Then, we can use the point-slope form of a linear equation to determine an equation of the tangent line.
First, let's find the derivative of the function f(x) = (15/4)x. The derivative of a function represents the rate at which the function is changing at any given point.
The derivative of f(x) = (15/4)x can be found by applying the power rule of differentiation, which states that the derivative of x^n (where n is a constant) is nx^(n-1). Since the constant 15/4 can be considered x^(1-0), we can apply the power rule.
The derivative of f(x) = (15/4)x is:
f'(x) = (15/4) * 1x^(1-1) = (15/4) * 1 = 15/4
Now that we have the derivative, let's evaluate it at the given point (1, 15/4) to find the slope. Since the slope represents the rate of change of the function at that point, evaluating the derivative at that point will give us the slope of the tangent line.
f'(1) = 15/4
So, the slope (m) of the tangent line is 15/4.
Now that we have the slope, we can determine the equation of the tangent line using the point-slope form of a linear equation, which is:
y - y1 = m(x - x1)
Substituting the given point (1, 15/4) and the slope we found (15/4) into the equation, we get:
y - (15/4) = (15/4)(x - 1)
Simplifying, we obtain:
y - (15/4) = (15/4)x - (15/4)
y = (15/4)x + (15/4) - (15/4)
y = (15/4)x
Therefore, the equation of the tangent line to the graph of f(x) = (15/4)x at the point (1, 15/4) is y = (15/4)x.