# Product Rule

The product rule is a fundamental concept in calculus used to compute the derivative of the product of two functions. If you have two functions, $f(x)$ and $g(x)$, the product rule helps you find the derivative of their product

$h(x) = f(x)g(x)$with respect to $x$. Mathematically, the product rule states that:

$h'(x) = f'(x)g(x) + f(x)g'(x)$where $h'(x)$ is the derivative of $h(x)$ with respect to $x$, $f'(x)$ is the derivative of $f(x)$ with respect to $x$, and $g'(x)$ is the derivative of $g(x)$ with respect to $x$.

## Examples

**Find the derivative of the product function $h(x) = (x^2 + 1)(3x - 4)$ with respect to $x$**

- Define the functions $f(x)$ and $g(x)$:

- Compute the derivatives:

- Apply the product rule:

In Sympy this looks like:

```
import sympy
x = sympy.symbols("x")
u_x = x**2 + 1
v_x = 3 * x - 4
u_prime = sympy.diff(u_x, x)
v_prime = sympy.diff(v_x, x)
w_prime = u_prime * v_x + u_x * v_prime
sympy.simplify(w_prime)
```

Tags: Mathematics