# How do you find the gradient of a function at a given point?

Sep 4, 2014

The gradient of a function is also known as the slope, and the slope (of a tangent) at a given point on a function is also known as the derivative.

To find the gradient, take the derivative of the function with respect to $x$, then substitute the x-coordinate of the point of interest in for the $x$ values in the derivative.

For example, if you want to know the gradient of the function $y = 4 {x}^{3} - 2 {x}^{2} + 7$ at the point $\left(1 , 9\right)$ we would do the following:

1. Take the derivative with respect to $x$:
$12 {x}^{2} - 4 x$

2. Substitute the x-coordinate $\left(x = 1\right)$ in for $x$:
gradient = $12 {\left(1\right)}^{2} - 4 \left(1\right) = 8$

So the gradient of the function at the point $\left(1 , 9\right)$ is $8$.