Question

Find the derivative of the algebraic function. $$ f(x)=\left(3 x^{3}+4 x\right)(x-5)(x+1) $$

Short answer

After simplifying the result, the final derivative of the function \(f(x) = (3x^3 + 4x)(x-5)(x+1)\) is obtained.

Step-by-step solution

Identifying the Functions

First, identify the three functions that make up the given function. Let's define them as follows: \(g(x) = 3x^3 + 4x\), \(h(x) = x-5\), and \(i(x) = x+1\).

Finding the Derivatives of the Functions

Next, using the power rule, find the derivatives of the functions: \(g'(x) = 9x^2 + 4\), \(h'(x) = 1\), and \(i'(x) = 1\). The power rule, \(d/dx[x^n] = nx^{n-1}\), states that the derivative of \(x^n\), with respect to x, is \(nx^{n-1}\). A quick reminder that the derivative of a constant is zero.

Applying the Product Rule

Now, apply the product rule: \(d/dx[uv] = u'v + uv'\). The derivative of the given function \(f(x) = g(x)h(x)i(x)\) is: \(f'(x) = g'(x)h(x)i(x) + g(x)h'(x)i(x) + g(x)h(x)i'(x)\). Substitute the corresponding functions and their derivatives.

Substituting the Functions and Derivatives

Now substitute the functions and their derivatives into the previous formula. This gives: \(f'(x) = (9x^2 + 4) * (x-5)(x+1) + (3x^3 + 4x) * 1 * (x+1) + (3x^3 + 4x) * (x-5) * 1\).

Simplifying the Result

Finally, simplify the resulting equation to obtain a clear derivative.