Instantaneous Rate of Change Calculator
This calculator will find the instantaneous rate of change of the given function at the given point, with steps shown.
Your Input
Find the instantaneous rate of change of $$$f{\left(x \right)} = x^{3} + 5 x^{2} + 7 x + 4$$$ at $$$x = 6$$$.
Solution
The instantaneous rate of change of the function $$$f{\left(x \right)}$$$ at the point $$$x = x_{0}$$$ is the derivative of the function $$$f{\left(x \right)}$$$ evaluated at the point $$$x = x_{0}$$$.
This means that we need to find the derivative of $$$x^{3} + 5 x^{2} + 7 x + 4$$$ and evaluate it at $$$x = 6$$$.
So, find the derivative of the function: $$$\frac{d}{dx} \left(x^{3} + 5 x^{2} + 7 x + 4\right) = \left(x + 1\right) \left(3 x + 7\right)$$$ (for steps, see derivative calculator).
Finally, evaluate the derivative at $$$x = 6$$$.
$$$\left(\frac{d}{dx} \left(x^{3} + 5 x^{2} + 7 x + 4\right)\right)|_{\left(x = 6\right)} = \left(\left(x + 1\right) \left(3 x + 7\right)\right)|_{\left(x = 6\right)} = 175$$$
Therefore, the instantaneous rate of change of $$$f{\left(x \right)} = x^{3} + 5 x^{2} + 7 x + 4$$$ at $$$x = 6$$$ is $$$175$$$.