Fundamentals of Differentiation

Which of the following expressions represents the derivative of a function $f(x)$?

If the derivative of a function $f(x)$ at $x=a$ is 5, what is the slope of the tangent line to $f(x)$ at $x=a$?

Find the equation of the tangent line to the curve $y = x^2 + 3x - 1$ at the point $(1, 3)$.

Find the derivative of $f(x) = 4x^5 - 3x^2 + 2x - 1$.

At which of the following points does the derivative of the function graphed below NOT exist?

Determine if the following piecewise function is differentiable at x=0: $f(x) = \begin{cases} x^2 + 1, & x < 0 \\ -x^2 + 1, & x \geq 0 \end{cases}$

Find the derivative of $f(x) = 5x^3 - 2x + 7$.

Find the derivative of $f(x) = 3x^2 + 2x - 1$ using the limit definition.

Suppose the tangent line to the curve $y = x^3 + ax^2$ at $x = 1$ passes through the point $(2, 0)$. Find the value of $a$.

What does the derivative of a function at a point represent?