If f(x)=4x^(5/2), find f'(4)
Use this to find the equation of the tangent line to the curve y=4x^(5/2) at the point (4,f(4)) . The equation of this tangent line can be written in the form y=mx+b where m is______ and where b is _____
To find the derivative of a function and evaluate it at a specific value, we can use the power rule for differentiation. The power rule states that for any function of the form f(x) = ax^n, the derivative is given by f'(x) = anx^(n-1).
In this case, f(x) = 4x^(5/2). To find f'(x), we can apply the power rule:
f'(x) = 4 * (5/2) * x^(5/2 - 1)
= 4 * (5/2) * x^(3/2)
= 10x^(3/2)
Now, to find f'(4), we substitute x = 4 into the equation:
f'(4) = 10 * 4^(3/2)
= 10 * 8
= 80
Therefore, f'(4) = 80.
To find the equation of the tangent line to the curve y = 4x^(5/2) at the point (4, f(4)), we can use the point-slope form of a linear equation, which is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope of the line.
We already have the point (x1, y1) = (4, f(4)). The slope, m, is given by f'(4) = 80.
Substituting these values into the equation, we have:
y - f(4) = 80(x - 4)
Simplifying, we get:
y = 80x - 320 + f(4)
Therefore, the equation of the tangent line is y = 80x - 320 + f(4). In this form, m = 80 and b = -320 + f(4). Since the value of f(4) is not given, we leave b as -320 + f(4).