Derivative Notation and Language

Type of Notation

Prime Notation:

Named because you use little prime tic marks.

Differential Notation:

Named for its use of what are known as differentials (dx, dy, dt)

Directions you will see that tell you to find the derivative.

With prime notation it will normally be in the language.

“Differentiate $f\left(x\right)$”

Differential notation uses notation to say everything.

$\frac{d}{\mathit{dx}}\left(\mathit{The\; Equation}\right)$

$\frac{d}{\mathit{dx}}(x^2+3x+9)$

The top of the notation is saying take the derivative of this equation, and the bottom is saying with respect to x (this is the variable you care about).

First Derivative

Taking the derivative once.

$f\prime \left(x\right),\mathit{ f}\prime \left(t\right),g\prime \left(x\right),\mathit{ y}\prime$

Read “f prime of x”

$\frac{\mathit{dy}}{\mathit{dx}}, \frac{\mathit{dy}}{\mathit{dt}}$

Read: “d…y…d…x”

Or “derivative of y with respect to x”

Second Derivative

Taking the derivative twice.

$f\prime \prime \left(x\right),\mathit{ f}\prime \prime \left(t\right),g\prime \prime \left(x\right)$

Read “f double prime of x”

$\frac{d^{2}y}{d{x}^2}, \frac{d^{2}y}{d{t}^2}$

Higher Order Derivative

Take the derivative as many times as they want you to. 3 times, 100 times, n-times

$f^{\left(3\right)}\left(x\right),f^{\left(100\right)}\left(x\right),f^{\left(n\right)}\left(x\right)$

$\frac{d^{3}y}{d{x}^3}, \frac{d^{100}y}{d{t}^{100}},\frac{d^{\left(n\right)}y}{d{t}^{\left(n\right)}}$

Find the derivative at x = 3 .

$f\prime \left(3\right)=$

$\frac{\mathit{dy}}{\mathit{dx}}|\begin{array}{c}\_\\ x=3\end{array}=$

Find the slope of the tangent line at x = 3 .

$f\prime \left(3\right)=$

$\frac{\mathit{dy}}{\mathit{dx}}|\begin{array}{c}\_\\ x=3\end{array}=$