calculate derivative of f(x)=5x (x-1)^4

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We have to find the derivative of f(x) = 5x * (x-1)^4

We can use the product rule here:

f(x) = 5x * (x-1)^4

f'(x) = (5x)' * (x-1)^4 + 5x * [(x-1)^4]'

=> f'(x) = 5* ( x - 1)^4 + 5x * 4*(x - 1)^3

=> f'(x) = 5(x - 1)^4 + 20x(x - 1)^3

The required derivative is f'(x) = 5*(x - 1)^4 + 20*x*(x - 1)^3

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We also could expand the binomial and then we'll differentiate each term of the polynomial.

(x-1)^4 = x^4 - 4x^3 + 6x^2 - 4x + 1 (1)

Now, we'll multiply (1) by 5x:

5x(x-1)^4 = 5x(x^4 - 4x^3 + 6x^2 - 4x + 1)

5x(x-1)^4 = 5x^5 - 20x^4 + 30x^3 - 20x^2 + 5x

We'll differentiate with respect to x and we'll get:

f'(x) = 25x^4 - 80x^3 + 90x^2 - 40x + 5

We'll factorize by 5:

f'(x) = 5(5x^4 - 16x^3 + 18x^2 - 8x + 1)

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