Best solutions to - "Find the derivatives of f(x)=32x4+2x3−5x2+9"

amanf

Answered question

2021-09-15

Find the derivatives of $f (x) = \frac{3}{2} x^{4} + 2 x^{3} - 5 x^{2} + 9$

Answer & Explanation

wornoutwomanC

Skilled2021-09-16Added 81 answers

Find the derivative of the following via implicit differentiation:
$\frac{d}{d x} (f (x)) = \frac{d}{d x} (9 - 5 x^{2} + 2 x^{3} + \frac{3 x^{4}}{2})$
Using the chain rule, $\frac{d}{d x} (f (x)) = \frac{d f (u)}{d u} \frac{d u}{d x}$ , where $u = x and \frac{d}{d u} (f (u)) = f^{'} (u)$ :
$(\frac{d}{d x} (x)) f^{'} (x) = \frac{d}{d x} (9 - 5 x^{2} + 2 x^{3} + \frac{3 x^{4}}{2})$
The derivative of x is 1:
$1 f^{'} (x) = \frac{d}{d x} (9 - 5 x^{2} + 2 x^{3} + \frac{3 x^{4}}{2})$
Differentiate the sum term by term and factor out constants:
$f^{'} (x) = \frac{d}{d x} (9) - 5 (\frac{d}{d x} (x^{2})) + 2 (\frac{d}{d x} (x^{3})) + \frac{3}{2} (\frac{d}{d x} (x^{4}))$
The derivative of 9 is zero:
$f^{'} (x) = - 5 (\frac{d}{d x} (x^{2})) + 2 (\frac{d}{d x} (x^{3})) + \frac{3}{2} (\frac{d}{d x} (x^{4})) + 0$
Simplify the expression:
$f^{'} (x) = - 5 (\frac{d}{d x} (x^{2})) + 2 (\frac{d}{d x} (x^{3})) + \frac{3}{2} (\frac{d}{d x} (x^{4}))$
Use the power rule, $\frac{d}{d x} (x^{n}) = n x^{n - 1}$ , where n = 2.
$\frac{d}{d x} (x^{2}) = 2 x :$
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