Explained: "Differentiate the following function using product rule. f(x)=(34x3−2x2+x4−6)(2x2−2x+4)"

untchick04tm

Answered question

2021-12-10

Differentiate the following function using product rule.

f (x) = (\frac{3}{4} x^{3} - 2 x^{2} + \frac{x}{4} - 6) (2 x^{2} - 2 x + 4)

Answer & Explanation

xandir307dc

Beginner2021-12-11Added 35 answers

Step 1
For functions given as product of functions f(x)g(x) the derivative using product rule is given as (f(x)g(x))'=f'(x)g(x)+f(x)g'(x). One property of derivatives that will be used is (cf(x))'=cf'(x). Here c is a constant.
Another property of derivatives that will be used is

\frac{d}{d x} x^{n} = n x^{n - 1}

. This is for

n \neq 0

.
Step 2
The given function is

f (x) = (\frac{3}{4} x^{3} - 2 x^{2} + \frac{x}{4} - 6) (2 x^{2} - 2 x + 4)

. Calculate the derivative using the product rule and other properties.

f (x) = (\frac{3}{4} x^{3} - 2 x^{2} + \frac{x}{4} - 6) (2 x^{2} - 2 x + 4)

f^{'} (x) = {(\frac{3}{4} x^{3} - 2 x^{2} + \frac{x}{4} - 6)}^{'} (2 x^{2} - 2 x + 4) + (\frac{3}{4} x^{3} - 2 x^{2} + \frac{x}{4} - 6) {(2 x^{2} - 2 x + 4)}^{'}

= (\frac{9 x^{2}}{4} - 4 x + \frac{1}{4}) (2 x^{2} - 2 x + 4) + (\frac{3}{4} x^{3} - 2 x^{2} + \frac{x}{4} - 6) (4 x - 2)

Hence, the derivative of given function is

(\frac{9 x^{2}}{4} - 4 x + \frac{1}{4}) (2 x^{2} - 2 x + 4) + (\frac{3}{4} x^{3} - 2 x^{2} + \frac{x}{4} - 6) (4 x - 2)

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