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The derivative of an exponent of a function. Differentiating power functions such as y = (2x+4)^3

Updated on September 30, 2011


There are basically two ways that you can use to differentiate exponents of a function. The first way (which is probably the easiest) is to apply this general formula:

If y = [f(x)]n then dy/dx = n.f´(x).[f(x)]n-1.

The second method is to use the chain rule as y = [f(x)]n is a function of a function. Make y = un and u = f(x). Differentiate both of these and multiply them together to give you dy/dx.

Let’s take a look at some examples of differentiating powers of a function.

Example 1

Differentiate y = (5x+4)3.

Let’s do the first method for differentiating exponents of a function:

f(x) = 5x + 2 so f´(x) = 5, and n = 3. All you need to do next is substitute these values into the formula above:

dy/dx = n.f´(x).[f(x)]n-1

dy/dx = 3×5×(5x+2)2

= 15(5x+2)2

Now try out the chain rule method:

y = u3 and u = 5x+4

dy/du = 3u2 and du/dx = 5

so dy/dx = 3u² × 5 = 15u2.

Finally, substitute u = 5x + 4 into the above so you have the answer in terms of x only:

dy/dx = 15(5x+2)2

Example 2

Differentiate y = (7x-2)4.

Let’s do the first method for differentiating exponents of a function:

f(x) = 7x-2 so f´(x) = 7, and n = 4. All you need to do next is substitute these values into the formula above:

dy/dx = n.f´(x).[f(x)]n-1

dy/dx = 4×7×(7x- 2)3

= 28(7x-2)3

Now try out the chain rule method:

y = u4 and u = 7x-2

dy/du = 4u3 and du/dx = 7

so dy/dx = 4u3 × 7 = 28u3.

Finally, substitute u = 5x + 4 into the above so you have the answer in terms of x only:

dy/dx = 28(7x-2)3

Example 3

Differentiate y = (2x5+ 3x2)-0.5.

Let’s do the first method for differentiating exponents of a function:

f(x) = 2x5+ 3x2 so f´(x) = 10x4 + 6x, and n = -0.5. All you need to do next is substitute these values into the formula above:

dy/dx = n.f´(x).[f(x)]n-1

dy/dx = -0.5 × (10x4 + 6x)×( 2x5+ 3x2)-1.5

= -0.5(10x4 + 6x)( 2x5+ 3x2)-1.5

Now try out the chain rule method:

y = u-0.5 and u = 2x5+ 3x2

dy/du = -0.5u-1.5 and du/dx = 10x4 + 6x

so dy/dx = -0.5u-1.5 × (10x4 + 6x)

= -0.5(10x4 + 6x) u-1.5

Finally, substitute 2x5+ 3x2 into the above so you have the answer in terms of x only:

dy/dx = -0.5(10x4 + 6x)( 2x5+ 3x2)-1.5

So that’s all there is to it if you have to differentiate exponents of a function. Remember that using the formula (method 1) is a much quicker method to use than the chain rule.

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