1. Derivatives
AP Physics C

2. Reminder
If we say that y is a function of x, for example y(x)=3x2, then the derivative of y is defined as:
𝑑𝑦 𝑑𝑥 = lim ∆𝑥⟶0 Δ𝑦 Δ𝑥
In other words, the derivative equals the slope of the tangent to the graph at any point on the graph.

3. Derivative of y=xn 𝑑𝑦 𝑑𝑥 =𝑛 𝑥 𝑛−1 𝑦= 𝑥 4 𝑦= 𝑥 2
𝑑𝑦 𝑑𝑥 =4 𝑥 4−1 =4 𝑥 3
𝑦= 𝑥 2
𝑑𝑦 𝑑𝑥 =2 𝑥 2−1 =2 𝑥 1 =2𝑥
𝑦= 𝑥 6
𝑑𝑦 𝑑𝑥 =6 𝑥 6−1 =6 𝑥 5

4. Derivative of y=Axn 𝑑𝑦 𝑑𝑥 =𝑛𝐴 𝑥 𝑛−1 𝑦=6 𝑥 4 𝑑𝑦 𝑑𝑥 =4∙6 𝑥 4−1 =24 𝑥 3
𝑦=5 𝑥 2
𝑑𝑦 𝑑𝑥 =2∙5 𝑥 2−1 =10 𝑥 1 =10𝑥
𝑦=8𝑥
𝑑𝑦 𝑑𝑥 =1∙8 𝑥 1−1 =8 𝑥 0 =8

5. Derivatives of a linear equation
Remember we just had 𝑦=8𝑥 gives us 𝑑𝑦 𝑑𝑥 =8
So the derivative of 𝑦=8𝑥 is just a number (8).
Why?
What shape is the graph of 𝑦=8𝑥?
It is straight line
What is the slope of that line (remember 𝑦=𝑚𝑥+𝑏 form)?
The slope is 8. We said derivative equals slope at a given point and this line has a slope of 8 everywhere.

6. Derivatives of a linear equation
What about derivative of a constant?
That is 𝑦=7 (which is a horizontal line).
Let’s follow the pattern: 𝑑𝑦 𝑑𝑥 =𝑛𝐴 𝑥 𝑛−1
𝑦=7 𝑥 0
So 𝑑𝑦 𝑑𝑥 =𝑛𝐴 𝑥 𝑛−1 is 0∙7 𝑥 0−1 =0∙7 𝑥 −1 =0
So the derivative of a constant number is zero.
Which it should be because the slope of a horizontal line is zero.

7. Derivatives of a linear equation
The derivative of a linear (first power) function is always a constant number.
That number is just the coefficient of the variable.
𝑦=8𝑥 gives us 𝑑𝑦 𝑑𝑥 =8 in the previous example.
If 𝑥=3𝑡 , what is 𝑑𝑥 𝑑𝑡 ?
𝑑𝑥 𝑑𝑡 =3
If position is given by the equation 𝑥 =5𝑡 , what is the velocity?
Velocity = 𝑑 𝑥 𝑑𝑡 =5, that is 5m/s, forward.

8. Derivatives of Polynomials
Each term in a polynomial can be treated separately as if it were its own function.
𝑑(4 𝑥 2 ) 𝑑𝑥 =8𝑥 and 𝑑(3𝑥) 𝑑𝑥 =3 and 𝑑(7) 𝑑𝑥 =0
What is 𝑑(4 𝑥 2 +3𝑥+7) 𝑑𝑥 ?
8𝑥+3+0
8𝑥+3

9. Derivatives of Polynomials
If 𝑥=9 𝑡 2 −3𝑡+5, what is 𝑑𝑥 𝑑𝑡 ?
𝑑𝑥 𝑑𝑡 =18𝑡−3
If 𝑣=10𝑡+4, what is 𝑑𝑣 𝑑𝑡 ?
𝑑𝑣 𝑑𝑡 =10