# 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.

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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 $175$.

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