The derivative function
So far you have calculated $f'(a)$ for a specific point $a$. But the same limit can be set up for any $x$: the result is no longer a number, but another function, $f'(x)$, which gives the derivative at each point of the domain.
The derivative function
Apply the definition of the derivative using the limit calculation $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$ to find the derivative function of…
Apply the definition of the derivative using the limit calculation $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$ to find the derivative function of $f(x) = 3x^2 - 5x + 2$.
Domain of differentiability
Consider the function defined using a modulus $f(x) = |x^2 - 4|$. Express $f(x)$ as a piecewise function and calculate its derivative function $f'(x)$. Prove…
Consider the function defined using a modulus $f(x) = |x^2 - 4|$.
Express $f(x)$ as a piecewise function and calculate its derivative function $f'(x)$. Prove analytically why the function is not differentiable at the corner points indicated on the graph.
The sign of $f'(x)$ tells the story of how $f$ increases or decreases
Observe the graph of the function $f(x)$ representing the trajectory of a projectile as a function of time. From the visual representation, determine the sign of the…
Observe the graph of the function $f(x)$ representing the trajectory of a projectile as a function of time.
From the visual representation, determine the sign of the derivative function $f'(x)$ at the x-coordinates $x = 1$, $x = 2$, and $x = 3$. Justify your answer based on the geometric concept of the derivative.
The following graph represents the derivative function $f'(x)$ of a continuous function $f(x)$ across all of $\mathbb{R}$. Observing exclusively the behaviour of…
The following graph represents the derivative function $f'(x)$ of a continuous function $f(x)$ across all of $\mathbb{R}$.
Observing exclusively the behaviour of $f'(x)$, determine the intervals where the original function $f(x)$ is increasing and where it is decreasing. At which x-coordinates will $f(x)$ have relative extrema?
Relative extrema: $f'(x) = 0$
Find the x-coordinates of the points on the curve $f(x) = e^{3x}(x^2 - 4x + 2)$ at which the tangent line is horizontal.
Find the x-coordinates of the points on the curve $f(x) = e^{3x}(x^2 - 4x + 2)$ at which the tangent line is horizontal.
The second derivative
Determine the second derivative, $f''(x)$, of the function $f(x) = x^4 \cdot \ln(x)$.
Determine the second derivative, $f''(x)$, of the function $f(x) = x^4 \cdot \ln(x)$.