Differentiating functions with multiple terms

In a nutshell

Functions can consist of multiple terms, added or subtracted together. To differentiate these functions, you apply the differentiation methods you know to each term individually. In other words, you can differentiate a function term by term. Recall that to differentiate $$ax^n$$ with respect to $$x$$, you multiply by the power, then subtract one from the power giving: $$nax^{n-1}$$.​

Functions with multiple terms

If $$y=f(x)\pm g(x)$$, then the derivative will be: 

​$$\boxed{\dfrac{\text{d}y}{\text{d}x}=f'(x)\pm g'(x)}$$​​

Example 1

Differentiate $$y=x^2+x^3$$ with respect to $$x$$.

In other words, you can say that $$f(x)=x^2$$ and $$g(x)=x^3$$. These can be differentiated in turn. Differentiating $$f(x)$$ and $$g(x)$$ each with respect to $$x$$ gives:

$$f'(x)=2x\\\space\\g'(x)=3x^2$$​​

Hence, the derivative is:

$$\underline{\dfrac{\text{d}y}{\text{d}x}=2x+3x^2}$$​​

Simplify where possible first

Some expressions don't initially look like the sum of multiple terms, but simplification can make them easier to differentiate.

Example 2

Differentiate $$y=\dfrac{3x-5x^2+4x^6}{x}$$ with respect to $$x$$.

Start by dividing by the $$x$$ (from the denominator):

$$y=3-5x+4x^5$$​​

Now differentiate each term one by one. 

Note: $$f(x)=3$$ differentiates to $$f'(x)=0$$.

$$\dfrac{\text{d}y}{\text{d}x}=0-5+20x^4$$​​

This is better written as:

$$\underline{\dfrac{\text{d}y}{\text{d}x}=20x^4-5}$$​​

Example 3

Differentiate $$y=x^2\left(3x-7-\dfrac1{x^3}\right)$$ with respect to $$x$$.

Expand the brackets first:

$$y=3x^3-7x^2-\dfrac1{x}$$​​

Also, re-express the $$\dfrac1x$$ with fractional power notation:

$$y=3x^3-7x^2-x^{-1}$$​​

Now differentiate each term:

$$\dfrac{\text{d}y}{\text{d}x}=9x^2-14x+x^{-2}$$​​

This could also be given as 

$$\underline{\dfrac{\text{d}y}{\text{d}x}=9x^2-14x+\dfrac1{x^2}}$$​​

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