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Derivatives of expressions

Derivatives of expressions

Are functions your friend yet? How about derivatives?

Did you know that, sometimes, a function can be represented as only the expression? Well, they can — and guess what else? That can make it easier find the derivative.

Let’s learn how!

What does it mean to find the derivative of an expression?

To find the derivative of an expression is essentially the same as finding the derivative of a function. Sometimes we can simplify finding the derivative by removing f(x) and just finding the derivative of the expression that’s left.

A derivative of a function is a rate of change of that function with respect to a change in variable.

The more you know: finding the first derivative of the function f(x) at x0 actually means finding the slope of the tangent line to the graph of the function at x0.

To simplify the process of differentiation, we’ll use differentiation rules rather than the definition of the derivative. We’ve got the list for you right here — and trust us, you’ll need it!

Constant multiple property of derivatives ddx(c×f(x))=c×ddx(f(x))
Sum rule for derivatives ddx(f(x)+g(x))=ddx(f(x))+ddx(g(x))
Difference rule for derivatives ddx(f(x)g(x))=ddx(f(x))ddx(g(x))
Product rule for derivatives ddx(f(x)×g(x))=ddx(f(x))×g(x)+f(x)×ddx(g(x))
Quotient rule for derivatives ddx(f(x)g(x))=ddx(f(x))×g(x)f(x)×ddx(g(x))(g(x))2
The Chain rule (fg)(x)=f(g(x))×g(x) or dydx=dydududx when y=f(u),u=g(x)
Derivative of the inverse function (f1)(x)=1f(f1(x))

Why is the derivative so useful?

Functions are endlessly important, which means, by extension, derivatives are, too! Because they tell us the rate of change, derivatives are super important for physicists, economists, and so many more people in our world (maybe even more than you realize!).

How to find the derivative of an expression

Okay, let’s talk about how to find the derivative of an expression. If you’ve already learned how to find the derivative of a function, this will probably look pretty familiar:

Example 1


Find the derivative of the expression:

ddx(x2+3x)

Use the differentiation rule ddx(f+g)=ddx(f)+ddx(g):

ddx(x2)+ddx(3x)

Next, use ddx(xn)=n×xn1:

2x1+ddx(3x)

Any expression raised to the power of 1 equals itself, so:

2x+ddx(3x)

Use ddx(a×x)=a:

2x+3

There we go! The derivative of the expression x2+3x is:

 

2x+3

See? Not so bad. Let’s do one more!

Example 2


Find the derivative of the expression:

ddx(lnx×x)

Use the differentiation rule ddx(f×g)=ddx(f)×g+f×ddx(g):

ddx(lnx)×x+lnx×ddx(x)

Use ddx(lnx)=1x:

1x×x+lnx×ddx(x)

Next, use ddx(x)=1:

1x×x+lnx×1

Simplify the expression:

1+lnx

We did it again! The derivative of an expression lnx×x is:

1+lnx

As you can see, it’s not so different from taking the derivative of a function. To review, here’s the process you should follow for finding the derivative of an expression:

Study summary

  1. Use the differentiation rules.
  2. Find the derivative of each term.
  3. Simplify the expression.

Do it yourself!

Want some more practice? Try your hand at these problems and let us know if you get stuck:

Take the derivative of an expression:

  1. ddx(ex+5x)
  2. ddx(ln(x)+5x)
  3. ddx(x41)
  4. ddx(x1x24)

Solutions:

  1. ex+5
  2. 1x+ln5×5x
  3. 2x3x41
  4. x2+2x4(x24)2

How did you do? Do you want to check your work? Scan the problem using your Photomath app to see detailed step-by-steps of the solving method.

Here’s a sneak peek of what you’ll see: