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


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.

More by this Author


Comments

No comments yet.

    0 of 8192 characters used
    Post Comment

    No HTML is allowed in comments, but URLs will be hyperlinked. Comments are not for promoting your articles or other sites.


    Click to Rate This Article
    working