#### Differential Arctan

Consider $f(x)= \dfrac{\pi}{2} -2 \arctan(1-2x)$ with domain $D_f=\mathbb{R}$. Knowing that the value of $df(x_0)=4 \ dx$, the value of $x_0$, is:

# Complements of differential calculus in real numbers

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#### Arctan inverse function derivative theorem

Consider $y=f(x)= -2 \arctan\left(2x\right)$ with domain $D_f=\mathbb{R}$. Applying the inverse function derivative theorem, the expression of $\dfrac{dx}{dy}$ is:

# Complements of differential calculus in real numbers

Training
13

#### Arctan inverse function derivative theorem

Consider $y=f(x)= -2 \arctan\left(2x\right)$ with domain $D_f=\mathbb{R}$. Applying the inverse function derivative theorem, the expression of $\dfrac{dy}{dx}$ is:

# Complements of differential calculus in real numbers

Training
13

#### Arctan inverse function derivative theorem

Consider $y=f(x)= \dfrac{\pi}{2} -2 \arctan(1-2x)$ with domain $D_f=\mathbb{R}$. Applying the inverse function derivative theorem, the expression of $\dfrac{dy}{dx}$ is:

# Complements of differential calculus in real numbers

Training
13

#### Derivation arctan

Let $\displaystyle f(x)= \dfrac{\pi}{2} -b \arctan(1-2x)$ with $b \in \mathbb{R}$, and let $f'(x)$ and $f''(x)$ be the first and the second derivative of the function $f$. Knowing that $\displaystyle f\left(\dfrac{1}{2}\right) \times f'\left(\dfrac{1}{2}\right) + f''\left(\frac{1}{2}\right) = 4 \pi$, then the value of $b$ is:

# Complements of differential calculus in real numbers

Training
13

#### Derivation arctan

Consider $f(x)= -2 \arctan\left(2x\right)$ with domain $D_f=\mathbb{R}$. Let $f'(x)$ be the first derivative of function $f(x)$, then $f'\left(-\dfrac{1}{2}\right)$ is:

# Complements of differential calculus in real numbers

Training
13