Power rule for derivatives

 
 
 
 
 

Other derivative rules, like power rule, are faster than using the definition of the derivative

At this point, we understand the idea of the derivative, and we know how to find it using the definition.

While the definition of the derivative can always be used to find the derivative of a function, it’s not usually the most efficient way of finding the derivative.

It’ll be faster for us to use the derivative rules we’re about to learn. In this lesson, we’ll look at the first of those derivative rules, which is the power rule.

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The power rule

The power rule lets us take the derivative of power functions. Power functions are things like x2x^2, 3x43x^4, 6x56x^5, etc.

Power rule tells us that, to take the derivative of a function like these ones, we just multiply the exponent by the coefficient, and then subtract 11 from the exponent.

Formally, power rule says that, for any function of the form axnax^n, the derivative will be

ddx(axn)=(an)xn1\frac{d}{dx}(ax^n)=(a\cdot n)x^{n-1}

For instance, to find the derivative of 3x43x^4, we’ll bring down the exponent of 44 to multiply it by the coefficient of 33, and we’ll subtract 11 from the exponent of 44. So the derivative would be

3(4)x413(4)x^{4-1}

12x312x^3

We can also use power rule to find the derivative of polynomials, which are combinations of power functions.

 
 

How to use power rule to take the derivative of any polynomial function


 
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Differentiating a polynomial function using the power rule for derivatives

Example

Find the derivative of the function.

f(x)=7x3+2x23xf(x)=7x^3+2x^2-3x

We can use power rule to take the derivative of the function one term at a time. We’ll apply the power rule to each term.

f(x)=7(3)x31+2(2)x213(1)x11f'(x)=7(3)x^{3-1}+2(2)x^{2-1}-3(1)x^{1-1}

f(x)=21x31+4x213x11f'(x)=21x^{3-1}+4x^{2-1}-3x^{1-1}

f(x)=21x2+4x13x0f'(x)=21x^2+4x^1-3x^0

f(x)=21x2+4x3(1)f'(x)=21x^2+4x-3(1)

f(x)=21x2+4x3f'(x)=21x^2+4x-3


We want to notice a couple of things about this last example. First, after applying power rule, we ended up with 4x14x^1 for the second term of the derivative. It’s not necessary to write an exponent when the exponent is 11; it’s implied. So 4x14x^1 can be written more simply as just 4x4x.

Second, we used power rule to take the derivative of the third term, 3x-3x. To apply power rule, we had to realize that 3x-3x is equivalent to 3x1-3x^1, so that we could use the exponent of 11. After applying power rule, like normal, to 3x1-3x^1, we got 3x0-3x^0. Anything raised to the 00 power is equal to 11, so x0x^0 turns into 11.

The takeaway here is that the derivative of any term where the exponent is 11, will be equal to the coefficient. So the derivative of 3x-3x is 3-3, the derivative of 7x7x is 77, and the derivative of xx is 11.

The derivative of a constant

Similarly, power rule tells us that the derivative of any constant will always be 00. In other words, the derivative of 3-3 is 00, the derivative of 77 is 00, and the derivative of 11 is 00.

As an example, take the constant 7-7. We can rewrite 7-7 as 7x0-7x^0, since x0x^0 is equivalent to 11, and multiplying 11 doesn’t change the value of the constant.  If we then use the power rule to take the derivative of the constant 7x0-7x^0, we get 7(0)x01-7(0)x^{0-1}. But because we now have 00 multiplying the constant, we get 00 for the value of the derivative.

Let’s do one more example where we apply the power rule to terms with different exponents.

Similarly, power rule tells us that the derivative of any constant will always be 0. In other words, the derivative of -3 is 0, the derivative of 7 is 0, and the derivative of 1 is 0.

Example

Find the derivative of the function.

f(x)=2x33x2+6x5f(x)=-2x^3-3x^2+6x-5

We’ll take the derivative one term at a time. We already know the derivative of 6x6x will be 66, and that the derivative of the constant 5-5 will be 00.

f(x)=2(3)x313(2)x21+60f'(x)=-2(3)x^{3-1}-3(2)x^{2-1}+6-0

f(x)=6x316x21+6f'(x)=-6x^{3-1}-6x^{2-1}+6

f(x)=6x26x1+6f'(x)=-6x^2-6x^1+6

f(x)=6x26x+6f'(x)=-6x^2-6x+6


Derivatives of combinations

Now that we’ve defined the power rule, and the rule for the derivative of a constant, let’s summarize the set of basic derivative rules.

Constant rule: ddx(a)=0\frac{d}{dx}(a)=0

Constant multiple rule: ddx(af(x))=af(x)\frac{d}{dx}(af(x))=af'(x)

Power rule: ddx(axn)=(an)xn1\frac{d}{dx}(ax^n)=(a\cdot n)x^{n-1}

Sum rule: ddx(f(x)+g(x))=f(x)+g(x)\frac{d}{dx}(f(x)+g(x))=f'(x)+g'(x)

Difference rule: ddx(f(x)g(x))=f(x)g(x)\frac{d}{dx}(f(x)-g(x))=f'(x)-g'(x)

We actually see all five of these rules used in the last example, where we differentiated f(x)=2x33x2+6x5f(x)=-2x^3-3x^2+6x-5. We use the sum and difference rules to take the derivative of the entire polynomial, one term at a time. We use the constant multiple rule and power rule to differentiate the first three terms, 2x33x2+6x-2x^3-3x^2+6x, and the constant rule to differentiate the last term, 5-5.

 
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