Question

Use the Second Fundamental Theorem of Calculus, if needed, to calculate each the derivatives expressed in Exercises 35–48.
$\frac{d}{dx}{\int }_0^{x^2}\cos t.dt$

Short answer

The derivative expression of $\frac{d}{dx}{\int }_0^{x^2}\cos t.dt$is $\cos x^2\left(2x\right)$.

Step-by-step solution

Step 1. Given Information.

The derivative:

$\frac{d}{dx}{\int }_0^{x^2}\cos t.dt$

Step 2. Second Fundamental theorem of calculus.

f is continuous on [a,b]for all $x\in [a,b]$, then$\frac{d}{dx}{\int }_e^{u\left(x\right)}f\left(t\right).dt=f\left(u\right(x\left)\right)u\text{'}\left(x\right)$

Step 3. Find the derivative expression.

By Second Fundamental theorem of calculus,

$$\begin{gathered} \frac{d}{dx}{\int }_0^{x^2}\cos t.dt=\cos x^2\left(x^2\right)\text{'} \\ =\cos x^2\left(2x\right) \end{gathered}$$