Instantaneous Rate of Change Calculator

Find the instantaneous rate of change of $$$f{\left(x \right)} = x^{3} + 5 x^{2} + 7 x + 4$$$ at $$$x = 6$$$.

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 $$$175$$$.

The instantaneous rate of $$$f{\left(x \right)} = x^{3} + 5 x^{2} + 7 x + 4$$$A at $$$x = 6$$$A is $$$175$$$A.