Let f(x) = 4x5/4 + 10x3/2 + 9x. Find the following.
(a)
f '(4)
f = 4x^(5/4) + 10x^(3/2)
f' = 4(5/4)x^(1/4) + 10(3/2)x^(1/2)
= 5∜x + 15√x
f'(4) = 5∜4 + 15√4 = 5√2 + 30
To find the derivative of the function f(x), you can use the power rule and the sum rule of differentiation. The power rule states that if you have a term of the form x^n, its derivative is n * x^(n-1). The sum rule states that if you have a sum of functions, the derivative of the sum is the sum of the derivatives of the individual functions.
Now let's differentiate each term of the function f(x):
f(x) = 4x^(5/4) + 10x^(3/2) + 9x
To find the derivative of the first term, 4x^(5/4), we apply the power rule. The derivative of this term is:
d/dx (4x^(5/4)) = (5/4) * 4 * x^(5/4 - 1) = 5x^(1/4)
Next, for the second term, 10x^(3/2), we again apply the power rule. The derivative is:
d/dx (10x^(3/2)) = (3/2) * 10 * x^(3/2 - 1) = 15x^(1/2)
Finally, for the third term, 9x, the power rule gives us:
d/dx (9x) = 9
Adding up these individual derivatives, we get:
f'(x) = 5x^(1/4) + 15x^(1/2) + 9
To find f'(4), we substitute x = 4 into the derivative expression:
f'(4) = 5(4)^(1/4) + 15(4)^(1/2) + 9
Now we can simplify the expression:
f'(4) = 5(2) + 15(2) + 9 = 10 + 30 + 9 = 49
Therefore, f'(4) = 49.