# 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 = u^{n} 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 = u^{3} and u = 5x+4

dy/du = 3u^{2} and du/dx = 5

so dy/dx = 3u² × 5 = 15u^{2}.

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 = u^{4} and u = 7x-2

dy/du = 4u^{3} and du/dx = 7

so dy/dx = 4u^{3} × 7 = 28u^{3}.

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 = (2x^{5}+ 3x^{2})^{-0.5}.

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

f(x) = 2x^{5}+ 3x^{2 }so f´(x) = 10x^{4} + 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 × (10x^{4} + 6x)×( 2x^{5}+ 3x^{2})^{-1.5}

= -0.5(10x^{4} + 6x)( 2x^{5}+ 3x^{2})^{-1.5}

Now try out the chain rule method:

y = u^{-0.5} and u = 2x^{5}+ 3x^{2}

dy/du = -0.5u^{-1.5} and du/dx = 10x^{4} + 6x

so dy/dx = -0.5u^{-1.5} × (10x^{4} + 6x)

= -0.5(10x^{4} + 6x) u^{-1.5}

Finally, substitute 2x^{5}+ 3x^{2 }into the above so you have the answer in terms of x only:

dy/dx = -0.5(10x^{4} + 6x)( 2x^{5}+ 3x^{2})^{-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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