All Tasks
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
Derivation arctan
Let $f$ and $g$ be two differentiable functions in their domain, such that:
$f(x)= \dfrac{\pi}{2} -2 \arctan(1-2x)$ and $g(x)=\left( x^2+1\right) f(x)$.
Considering that $g'(x)$ is the first derivative of function $g(x)$, then $g'\left(\dfrac{1}{2}\right)$ is:
# Complements of differential calculus in real numbers
Training
13