How to Find the Derivative of a Function Using the Product Rule in Mathematics

find derivative of f(x)= x(x+1)?

To find the derivative of the function f(x) = x(x + 1), we can use the product rule

To find the derivative of the function f(x) = x(x + 1), we can use the product rule. The product rule states that if we have two functions u(x) and v(x), the derivative of their product u(x)v(x) with respect to x is given by:

(u(x)v(x))’ = u'(x)v(x) + u(x)v'(x)

For our function f(x) = x(x + 1), we can consider u(x) = x and v(x) = (x + 1). Now let us find the derivatives of u(x) and v(x).

Derivative of u(x) = x:
Since u(x) is a linear function, the derivative is simply the coefficient of x, which is 1.

u'(x) = 1

Derivative of v(x) = (x + 1):
Using the power rule, we can take the derivative of (x + 1) as if it were x^n:
d/dx (x + 1) = 1^n * x^(n-1) = 1 * x^0 = 1

v'(x) = 1

Now, let’s use the product rule to find the derivative of f(x):

f'(x) = u'(x)v(x) + u(x)v'(x)
= 1 * (x + 1) + x * 1
= x + 1 + x
= 2x + 1

Therefore, the derivative of f(x) = x(x + 1) is f'(x) = 2x + 1.

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