Questions

$$\frac{d}{dx}(e)=\ ?$$

$$\frac{d}{dx}\,\int_{0}^{1} x\,dx=\ ?$$

$$\cos\!\Big(\,\frac{d}{dx}\int_{a}^{b} f(x)\,dx\Big)=\ ?$$

$$\frac{d}{dx}\,\int_{0}^{1} x\,dt=\ ?$$

Solutions

1) $e$ is a constant, not $e^{x}$.

$$\frac{d}{dx}(e)=0.$$

2) A definite integral with fixed bounds is a constant.

$$\int_{0}^{1} x\,dx=\tfrac12 \quad\Rightarrow\quad \frac{d}{dx}(\tfrac12)=0.$$

3) Inside the cosine, $\dfrac{d}{dx}\int_{a}^{b} f(x)\,dx=0$.

$$\cos(0)=1.$$

4) Here the integral is with respect to $t$, so $x$ is a constant during the integration.

$$\int_{0}^{1} x\,dt = x\!\int_{0}^{1} dt = x \quad\Rightarrow\quad \frac{d}{dx}(x)=1.$$