# Find d/dx?

Feb 7, 2018

The answer is $3 {\cos}^{2} \left(3 x\right)$

#### Explanation:

Where $a$ is a constant value, we know that:

$\frac{d}{\mathrm{dx}} {\int}_{a}^{x} f \left(t\right) \mathrm{dt} = f \left(x\right)$

Therefore, using the chain rule, we can say that:

$\frac{d}{\mathrm{dx}} {\int}_{a}^{u} f \left(t\right) \mathrm{dt} = f \left(u\right) \cdot \frac{\mathrm{du}}{\mathrm{dx}}$

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In this case, we can let $a = 1$, $u = 3 x$, and $f \left(t\right) = {\cos}^{2} \left(t\right)$.

Therefore:

$\frac{d}{\mathrm{dx}} {\int}_{1}^{3 x} {\cos}^{2} \left(t\right) \mathrm{dt} = {\cos}^{2} \left(3 x\right) \cdot \frac{d}{\mathrm{dx}} \left(3 x\right)$

$= {\cos}^{2} \left(3 x\right) \cdot 3$

$= 3 {\cos}^{2} \left(3 x\right)$