# Parametric Form of the Derivative

## First Derivative

Proof

## Second and Higher Degree Derivatives

The second derivative is the *derivative of the first derivative / derivative of x *, so it's:

# Example

Second derivative = $$\dfrac{\frac{d}{dt}t^\frac{3}{2}}{\dfrac{1}{2\sqrt{t}}} = \dfrac{\frac{3}{2}\sqrt{t}}{\dfrac{1}{2\sqrt{t}}} = 3t = 3(4) = \boxed{12}$$ So, the graph is concave up at (2,3), and its slope is 8.

x = √t, y = 1/4(t^{2} - 4), t≥0

Find the derivative and second derivative at (2,3)

dy/dx = (dy/dt)/(dx/dt) =

## (t/2) /(1/(2√t) = t√t = t^{3/2}

Derivative at (2,3): x = 2, t = 4, 4^{3/2} = **8**

Second derivative = d/dt [t^{3/2}]/(dx/dt) = (3/2)√t/(1/(2√t)) = 3t

When t = 4, 3t = 12.

So, the graph is concave up at (2,3), and the slope is 8.

# Speed

Sometimes, a problem might ask what the speed of a set of parametric equations is.

## Example

# Velocity

Sometimes, a problem might ask about the velocity.

## Example (Same as before, just with velocity)

# Finding a tangent line

This is the same as finding the tangent to any other curve. You create a line containing that point and the slope. You use point-slope form, so if the point is (a,b), you find what t is and you do y = b + g'(t)/f'(t)(x-a).