Let f(x) = 4x5/4 + 10x3/2 + 9x. Find the following.
(a) f '(0) =
(b) f '(16) =
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To find the derivative of the given function f(x), we can use the power rule and the sum rule for differentiation. The power rule states that if f(x) = ax^n, then f'(x) = nax^(n-1), and the sum rule states that if f(x) = g(x) + h(x), then f'(x) = g'(x) + h'(x).
Given: f(x) = 4x^(5/4) + 10x^(3/2) + 9x
(a) To find f'(0), we need to find the derivative of f(x) and evaluate it at x = 0. Let's differentiate each term of f(x) step by step:
For the first term, 4x^(5/4), we apply the power rule:
The derivative of 4x^(5/4) is (5/4) * 4 * x^(5/4 - 1) = 5x^(1/4).
Similarly, for the second term, 10x^(3/2), we apply the power rule:
The derivative of 10x^(3/2) is (3/2) * 10 * x^(3/2 - 1) = 15x^(1/2).
Lastly, for the third term, 9x, the derivative is simply 9, as the derivative of a constant multiplied by x is just the constant.
Now let's take the sum of these derivatives:
f'(x) = 5x^(1/4) + 15x^(1/2) + 9
To find f'(0), we substitute x = 0 into this expression for the derivative:
f'(0) = 5(0)^(1/4) + 15(0)^(1/2) + 9 = 0 + 0 + 9 = 9
Therefore, f'(0) = 9.
(b) To find f'(16), we need to evaluate the derivative at x = 16. Using the derivative we found earlier:
f'(x) = 5x^(1/4) + 15x^(1/2) + 9
Substituting x = 16 into this expression for the derivative:
f'(16) = 5(16)^(1/4) + 15(16)^(1/2) + 9
Simplifying this expression will give us the value of f'(16).
f'(x) = 5x^1/4 + 15x^1/2 + 9
now plug in 0 and 16 for x