8.9: Constant Derivatives and the Power Rule
The power rule is a fantastic "shortcut" for finding the derivatives of basic polynomials. Between the power rule and the basic definition of the derivative of a constant, a great number of polynomial derivatives can be identified with little effort  often in your head!
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 Khan Academy: Calculus: Derivatives 3
Guidance
In this lesson, we will develop formulas and theorems that will calculate derivatives in more efficient and quick ways. Look for these theorems in boxes throughout the lesson.
The Derivative of a Constant
Theorem: If
Proof: 

Theorem: If 

The Power Rule
Theorem: (The Power Rule) If n is a positive integer, then for all real values of x


Example A
Find
Solution

If
f(x)=16 for allx , thenf′(x)=0 for allx 
We can also write
ddx16=0
Example B
Find the derivative of
Solution

ddx[4x3] ..... Restate the function 
4ddx[x3] ..... Apply the Commutative Law 
4[3x2] ..... Apply the Power Rule 
12x2 ..... Simplify
Example C
Find the derivative of
Solution

ddx[−2x4] ..... Restate 
ddx[−2x−4] ..... Rules of exponents 
−2ddx[x−4] ..... By the Commutative law 
−2[−4x−4−1] ..... Apply the Power Rule 
−2[−4x−5] ..... Simplify 
8x−5 ..... Simplify again 
8x5 ..... Use rules of exponents
Vocabulary
A theorem is a statement accepted to be true based on a series of reasoned statements already accepted to be true. In the context of this lesson, a theorem is a rule that allows a quick calculation of the derivative of functions of different types.
A proof is a series of true statements leading to the acceptance of truth of a more complex statement.
Guided Practice
Questions
Find the derivatives of:
1)
2)
3)
4)
Solutions
1) By the power rule:

If
f(x)=x3 thenf(x)=(3)x3−2=3x2
2) Special application of the power rule:

ddx[x]=1x1−1=x0=1
3) Restate the function:

Using rules of exponents (from Algebra):
ddx[x1/2] 
Apply the Power Rule:
12x1/2−1 
Simplify:
12x−1/2 
Rules of exponents:
12x1/2 
Simplify:
12x√
4) Restate the function:

Rules of exponents:
ddx[x−3] 
Power Rule:
−3x−3−1 
Simplify:
−3x−4 
Rules of exponents:
−3x4
Practice
 State the Power Rule.
Find the derivative:

y=5x7 
y=−3x 
f(x)=13x+43  \begin{align*}y = x^4  2x^3  5\sqrt{x} + 10\end{align*}
 \begin{align*}y = (5x^2  3)^2\end{align*}
 given \begin{align*}y(x)= x^{4\pi^2}\end{align*} when \begin{align*} x = 1\end{align*}
 y(x) = 5</math>
 given \begin{align*}u(x)= x^{5\pi^3}\end{align*} what is \begin{align*} u'(2)\end{align*}
 \begin{align*} y = \frac{1}{5}\end{align*} when \begin{align*} x = 4 \end{align*}
 given \begin{align*}d(x)= x^{0.37}\end{align*} what is \begin{align*} d'(1)\end{align*}
 \begin{align*} g(x) = x^{3}\end{align*}
 \begin{align*}u(x) = x^{0.096}\end{align*}
 \begin{align*}k(x) = x{0.49}\end{align*}
 \begin{align*} y = x^{5\pi^3}\end{align*}
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derivative
The derivative of a function is the slope of the line tangent to the function at a given point on the graph. Notations for derivative include , , , and \frac{df(x)}{dx}.proof
A proof is a series of true statements leading to the acceptance of truth of a more complex statement.theorem
A theorem is a statement that can be proven true using postulates, definitions, and other theorems that have already been proven.Image Attributions
Here you will learn how to quickly identify the derivatives of constant terms, and you will explore the use of the power rule for finding the derivatives of higherorder functions.