What Is a Derivative in Calculus? | Outlier (2024)

In This Article

  1. What Is a Derivative?

  2. What Is Slope and Tangent?

  3. What Are the Derivative Formulas?

  4. Examples of How To Find the Derivative of a Function

  5. Where To Practice Derivatives

Differentiation is one of the most fundamental operations in calculus. Knowing how to find the derivative of a function will open up many doors in calculus.

What Is a Derivative?

Derivatives measure rates of change. More specifically, derivatives measure instantaneous rates of change at a point. The instantaneous rate of change of the function at a point is equal to the slope of the tangent line at that point.

The first derivative of a function fff at some given point aaa is denoted by f(a)f’(a)f(a). This expression is read aloud as “the derivative of fff evaluated at aaa” or “fff prime at aaa.”

The expression f(x)f’(x)f(x) is the notation used to denote the general derivative function of fff. We can also use Leibniz’s notation dydx\frac{dy}{dx}dxdy to denote the derivative function. We can plug x=ax = ax=a into f(x)f’(x)f(x) to determine the instantaneous rate of change of fff with respect to xxx at any point aaa on the curve.

The derivative of fff at xxx equals the limit of the average rate of change of fff over the interval [x,x+Δx][x, x +\Delta{x}][x,x+Δx] as Δx\Delta{x}Δx approaches 0, where Δx\Delta{x}Δx represents a change in xxx:

f(x)=limΔx0f(x+Δx)f(x)Δx=Lf’(x) = \mathop{\lim }\limits_{\Delta{x} \to 0} \frac{{f\left( {x + \Delta{x} } \right) - f\left( x\right)}}{\Delta{x} }=Lf(x)=Δx0limΔxf(x+Δx)f(x)=L

Here are the 3 steps to calculate a derivative using this definition:

  1. Substitute your function into the limit definition of a derivative formula.

  2. Simplify.

  3. Evaluate the resulting limit.

Example of How To Calculate a Derivative

Let’s do a very simple example together. Find the derivative of f(x)=3xf(x) = 3xf(x)=3x using the limit definition and the steps given above.

Step 1

The first step is to substitute f(x)=3xf(x) = 3xf(x)=3x into the limit definition of a derivative. The trick to this step is to substitute the variable xxx with the expression (x+Δx)(x + \Delta{x})(x+Δx) wherever xxx appears in f(x)=3xf(x) = 3xf(x)=3x.

f(x)=limΔx0f(x+Δx)f(x)Δxf’(x)= \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{{f\left( {x + \Delta{x} } \right) - f\left( x \right)}}{\Delta{x} }f(x)=Δx0limΔxf(x+Δx)f(x)

=limΔx03(x+Δx)3xΔx= \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{3(x + \Delta{x}) - 3x}{\Delta{x}}=Δx0limΔx3(x+Δx)3x

Step 2

The next step is to simplify. We’ll do this by expanding the numerator and combining like terms. Then, we can divide by Δx\Delta{x}Δx.

f(x)=limΔx03(x+Δx)3xΔxf’(x)= \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{3(x + \Delta{x}) - 3x}{\Delta{x}}f(x)=Δx0limΔx3(x+Δx)3x

=limΔx03x+3Δx3xΔx= \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{3x + 3\Delta{x} - 3x}{\Delta{x}}=Δx0limΔx3x+x3x

=limΔx03ΔxΔx= \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{3\Delta{x}}{\Delta{x}}=Δx0limΔxx

=limΔx03= \mathop {\lim }\limits_{\Delta{x} \to 0} 3=Δx0lim3

Step 3

Finally, we can evaluate the limit.

f(x)=limΔx03f’(x)= \mathop {\lim }\limits_{\Delta{x} \to 0} 3f(x)=Δx0lim3

f(x)=3f’(x)= 3f(x)=3

Using these steps, we’ve shown that the derivative function of f(x)=3xf(x) = 3xf(x)=3x is 3.

Since this is a constant function, the derivative of f(x)f(x)f(x) at any point is 3.

What Are Slope and Tangent?

In the above section, we learned the limit definition of a derivative:

f(x)=limΔx0f(x+Δx)f(x)Δx=Lf’(x) =\mathop{\lim }\limits_{\Delta{x} \to 0} \frac{{f\left( {x + \Delta{x} } \right) - f\left( x\right)}}{\Delta{x} }=Lf(x)=Δx0limΔxf(x+Δx)f(x)=L

But what does this equation mean, and how is it derived? We’ll answer this question in this section.

The slope of a straight line through (a,f(a))(a, f(a))(a,f(a)) and (b,f(b))(b, f(b))(b,f(b)) is equal to the change in yyy divided by the change in xxx:

Slope=ΔyΔx=f(b)f(a)ba\text{Slope } = \frac{\Delta y}{\Delta x} = \frac{f(b) - f(a)}{b-a}Slope=ΔxΔy=baf(b)f(a)

But what about curves that aren’t straight? For non-linear functions, we can instead find the slope of the curve at a specific point. The slope of a line at a specific point on a curve is called the slope of the tangent line. This value is equal to the instantaneous rate of change, or derivative, at that point.

The tangent line to a function at a specific point is a line that just barely touches the function at that point.

Take a look at the graph below, where the tangent line to the red curve f(x)=lnxf(x) = -\ln{x}f(x)=lnx at (1,0)(1,0)(1,0) is already graphed for us. The blue line represents this tangent line and has the equation f(x)=x+1f(x) = -x + 1f(x)=x+1. The blue line just barely touches the red line at the point (1,0)(1, 0)(1,0).

What Is a Derivative in Calculus? | Outlier (1)

Observe that the tangent line f(x)=x+1f(x) = -x + 1f(x)=x+1 is given in slope-intercept form. Slope-intercept form is f(x)=mx+bf(x) = mx +bf(x)=mx+b, where mmm is the slope. So, the slope of the tangent line f(x)=x+1f(x) = -x + 1f(x)=x+1 is -1. Since the derivative of a function at a point is equal to the slope of the tangent line at that point, this tells us that the derivative of f(x)=ln(x)f(x) = \ln{(x)}f(x)=ln(x) at the point (1,0)(1, 0)(1,0) is -1. We can also write f(1)=1f’(1) = -1f(1)=1.

Now we’ve learned that the slope of a curve at a specific point on a curve is called the slope of the tangent line. There’s another type of slope that is helpful to us when defining derivatives: the slope of the secant line. The slope between two separate points on a curve is called the slope of the secant line, which is also called the average rate of change.

You’re already familiar with the equation for this; it’s the same as the formula for the slope of a straight line!

AverageRateofChange=ΔyΔx=f(b)f(a)ba\text{Average Rate of Change} = \frac{\Delta{y}}{\Delta{x}} = \frac{f(b)-f(a)}{b-a}AverageRateofChange=ΔxΔy=baf(b)f(a)

Over the interval [x,x+Δx][x, x +\Delta{x}][x,x+Δx], this equation is equal to:

AverageRateofChange=f(x+Δx)f(x)Δx\text{Average Rate of Change} = \frac{f(x + \Delta{x})-f(x)}{\Delta{x}}AverageRateofChange=Δxf(x+Δx)f(x)

If we make Δx\Delta{x}Δx become closer and closer to 0 in the above equation of the secant line, we get closer and closer to finding the instantaneous rate of change of a function at xxx.

So, to find the instantaneous rate of change of a function at xxx, we can take the limit of the average rate of change of fff over the interval [x,x+Δx][x, x +\Delta{x}][x,x+Δx] as Δx\Delta{x}Δx approaches 0. This limit is the formal derivative definition formula:

f(x)=limΔx0f(x+Δx)f(x)Δx=Lf’(x) =\mathop{\lim }\limits_{\Delta{x} \to 0} \frac{{f\left( {x + \Delta{x} } \right) - f\left( x\right)}}{\Delta{x} }=Lf(x)=Δx0limΔxf(x+Δx)f(x)=L

If LLL exists, then fff is differentiable and LLL is the derivative of the function fff at xxx.

We can compute higher order derivatives as well. If the first derivative of a function is differentiable, we can also determine its second derivative. The second derivative is simply the derivative of the first derivative.

What Are the Derivative Formulas?

Once you understand the definition of a derivative, you can begin to become familiar with the most common derivative formulas. While it’s important to understand how to derive the limit definition of a derivative, these rules offer you some shortcuts to computing derivatives. These formulas allow you to calculate derivatives much faster than using the limit definition of a derivative.

Here are the most frequently used derivative equations. If you forget any of these rules during an exam, you can always rely on the limit definition to calculate the derivative.

Constant Rule

ddxc=0\frac{d}{dx}c = 0dxdc=0

Power Rule

ddx(xn)=nxn1\frac{d}{dx}(x^n) = nx^{n-1}dxd(xn)=nxn1

Special Case of the Power Rule (where n=1): ddx(x)=1\frac d{dx}(x)=1dxd(x)=1

Constant Multiple Rule

ddx(cf(x))=cf(x)\frac d{dx}(c\cdot f(x))=c\cdot f'(x)dxd(cf(x))=cf(x)

Chain Rule or Composite Functions Rule

ddxf(g(x))=f(g(x))g(x)\frac{d}{dx}f(g(x)) = f’(g(x))g’(x)dxdf(g(x))=f(g(x))g(x)

Product Rule

ddx[f(x)g(x)]=f(x)g(x)+f(x)g(x)\frac{d}{dx}[f(x) \cdot g(x)] = f’(x) \cdot g(x) + f(x)\cdot g’(x)dxd[f(x)g(x)]=f(x)g(x)+f(x)g(x)

Quotient Rule

ddx[f(x)g(x)]=g(x)f(x)f(x)g(x)(g(x))2\frac{d}{dx}[\frac{f(x)}{g(x)}] = \frac{g(x)f’(x)-f(x)g’(x)}{(g(x))^2}dxd[g(x)f(x)]=(g(x))2g(x)f(x)f(x)g(x)

Sum or Difference Rule

ddx[f(x)±g(x)]=f(x)±g(x)\frac{d}{dx}[f(x) \pm g(x)] = f’(x) \pm g’(x)dxd[f(x)±g(x)]=f(x)±g(x)

Trigonometry Rules

ddx(sin(x))=cos(x)\frac{d}{dx}(\sin{(x)}) = \cos{(x)}dxd(sin(x))=cos(x)

ddx(cos(x))=sin(x)\frac{d}{dx}(\cos{(x)}) = -\sin{(x)}dxd(cos(x))=sin(x)

ddx(tan(x))=sec2(x)\frac{d}{dx}(\tan{(x)}) = \sec ^2 (x)dxd(tan(x))=sec2(x)

Rules for Logarithmic Functions and Exponential Functions

ddx(lnx)=1x\frac{d}{dx} (\ln{x}) = \frac{1}{x}dxd(lnx)=x1

ddx(ex)=ex\frac{d}{dx}(e^x) = e^xdxd(ex)=ex

Dr. Tim Chartier highlights two of these rules, the Product Rule and Quotient Rule, as game changers:

He also goes over in this video the Constant Rule, Power Rule, and Sum Rule with examples:

Examples of How To Find the Derivative of a Function

Let’s try a few derivative examples together.

Finding a Derivative Example 1

Find the derivative of f(x)=4x2f(x) = 4x^2f(x)=4x2 using the limit definition of a derivative.

Solution

We’ll follow the three steps listed in the first section.

Step 1

Substituting our function f(x)=4x2f(x) = 4x^2f(x)=4x2 into the limit definition of a derivative, we get:

f(x)=limΔx0f(x+Δx)f(x)Δxf’(x)= \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{{f\left( {x + \Delta{x} } \right) - f\left( x \right)}}{\Delta{x} }f(x)=Δx0limΔxf(x+Δx)f(x)

=limΔx04(x+Δx)24x2Δx= \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{4(x + \Delta{x})^2 - 4x^2}{\Delta{x}}=Δx0limΔx4(x+Δx)24x2

Step 2

The next step is to simplify. We’ll do this by expanding the numerator and combining like terms. Then, we can divide by Δx\Delta{x}Δx.

f(x)=limΔx04(x2+2xΔx+Δx2)4x2Δxf’(x)= \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{4(x^2 + 2x\Delta{x} + \Delta{x}^2) - 4x^2}{\Delta{x}}f(x)=Δx0limΔx4(x2+2xΔx+Δx2)4x2

=limΔx04x2+8xΔx+4Δx24x2Δx= \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{4x^2 + 8x\Delta{x} + 4\Delta{x}^2 - 4x^2}{\Delta{x}}=Δx0limΔx4x2+8xΔx+x24x2

=limΔx08xΔx+4Δx2Δx= \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{8x\Delta{x} + 4\Delta{x}^2}{\Delta{x}}=Δx0limΔx8xΔx+x2

=limΔx08x+4Δx= \mathop {\lim }\limits_{\Delta{x} \to 0}8x + 4\Delta{x}=Δx0lim8x+x

Step 3

Now, we can evaluate the limit as Δx\Delta{x}Δx approaches 0. Polynomial functions are always continuous, so we can substitute Δx=0\Delta{x} = 0Δx=0 into the function.

f(x)=limΔx08x+4Δxf’(x)= \mathop {\lim }\limits_{\Delta{x} \to 0}8x + 4\Delta{x}f(x)=Δx0lim8x+x

f(x)=8x+4(0)f’(x) = 8x + 4(0)f(x)=8x+4(0)

f(x)=8xf’(x) = 8xf(x)=8x

Finding a Derivative Example 2

Let f(x)=7x3sin(3x)f(x) = 7x^3 - \sin{(3x)}f(x)=7x3sin(3x). Find f(x)f’(x)f(x) using the derivative rules.

Solution

We have the difference of two terms, so we can use the Difference Rule, which states that the derivative of a difference of functions is equal to the difference of their derivatives. To find the derivative of the first term, we can use the Power Rule and the Constant Multiple Rule:

ddx[7x3]=37x31=21x2\frac{d}{dx}[7x^3] = 3 \cdot 7 x^{3-1} = 21x^2dxd[7x3]=37x31=21x2

The second term is a composition of function. So, to find the derivative of the second term, we can use the Chain Rule. We’ll also need to use the sine rule of the trigonometry rules. The Chain Rule states that the derivative of a composition of functions is equal to the derivative of the outside function, multiplied by the derivative of the inside function:

dydxsin(3x)=cos(3x)3=3cos(3x)\frac{dy}{dx} \sin{(3x)} = \cos{(3x)} \cdot 3 = 3\cos{(3x)}dxdysin(3x)=cos(3x)3=3cos(3x)

Now, we can take the difference of these two derivatives to find the derivative of f(x)=7x3sin(3x)f(x) = 7x^3 - \sin{(3x)}f(x)=7x3sin(3x):

f(x)=21x23cos(3x)f’(x) = 21x^2 - 3\cos{(3x)}f(x)=21x23cos(3x)

Finding a Derivative Example 3

Let f(x)=cos(2x)2xf(x) = \frac{\cos{(2x)}}{2x}f(x)=2xcos(2x). Find f(x)f’(x)f(x) using the derivative rules.

Solution

We have a quotient of functions. So, we can use the Quotient Rule. In the numerator, we’ll need the cosine rule of the trigonometry rules, as well as the Chain Rule. In the denominator, we’ll use the Power Rule.

ddx[f(x)g(x)]=g(x)f(x)f(x)g(x)(g(x))2\frac{d}{dx}[\frac{f(x)}{g(x)}] = \frac{g(x)f’(x)-f(x)g’(x)}{(g(x))^2}dxd[g(x)f(x)]=(g(x))2g(x)f(x)f(x)g(x)

f(x)=2x[sin(2x)2]cos(2x)2(2x)2f’(x) = \frac{2x \cdot [-\sin{(2x)} \cdot 2] - \cos{(2x)} \cdot 2}{(2x)^2}f(x)=(2x)22x[sin(2x)2]cos(2x)2

f(x)=4xsin(2x)2cos(2x)4x2f’(x) = \frac{-4x\sin{(2x)} - 2\cos{(2x)}}{4x^2}f(x)=4x24xsin(2x)2cos(2x)

f(x)=sin(2x)xcos(2x)2x2f’(x) = \frac{-\sin{(2x)}}{x} - \frac{\cos{(2x)}}{2x^2}f(x)=xsin(2x)2x2cos(2x)

Where To Practice Derivatives

Outlier is a great resource for improving your mastery of derivatives. Once you’ve solidified your understanding of the derivative, Outlier’s calculus course is a fantastic way to expand your mathematical toolbox and apply your differentiation skills to other areas of differential calculus.

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I'm an expert in calculus with a deep understanding of derivatives. Let me demonstrate my expertise by providing information related to the concepts discussed in the article:

What Is a Derivative?

Derivatives measure rates of change and represent instantaneous rates of change at a specific point. The first derivative of a function at a given point is denoted as f'(a), and the general notation for the derivative function is f'(x) or dy/dx. The derivative is calculated using the limit definition, which involves substituting the function into the limit definition formula, simplifying, and then evaluating the limit.

What Is Slope and Tangent?

The slope of a straight line is defined as the change in y divided by the change in x. For curves, the slope of the tangent line at a specific point is equal to the instantaneous rate of change or derivative at that point. The tangent line touches the function at that point.

What Are the Derivative Formulas?

The article introduces various derivative formulas that provide shortcuts for calculating derivatives:

  • Constant Rule: The derivative of a constant is zero.
  • Power Rule: The derivative of x^n is n*x^(n-1).
  • Constant Multiple Rule: The derivative of a constant times a function is the constant times the derivative of the function.
  • Chain Rule: The derivative of a composite function is the derivative of the outer function multiplied by the derivative of the inner function.
  • Product Rule: The derivative of the product of two functions is the derivative of the first times the second plus the first times the derivative of the second.
  • Quotient Rule: The derivative of the quotient of two functions is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator.
  • Sum or Difference Rule: The derivative of the sum or difference of two functions is the sum or difference of their derivatives.

Examples of How To Find the Derivative of a Function

The article provides step-by-step examples of finding derivatives using the limit definition. It demonstrates the process with a simple function (f(x) = 3x) and a more complex one (f(x) = 4x^2). Additionally, it showcases finding the derivative of a function involving trigonometric and exponential components.

Where To Practice Derivatives

Outlier is recommended as a resource for practicing derivatives. Outlier offers interactive calculus courses taught by renowned math professors. The courses include question sets, quizzes, and a digital textbook for active learning. Credits earned from Outlier courses can be transferred, and flexible exam windows are provided.

This information provides a comprehensive overview of the concepts related to derivatives discussed in the article. If you have any specific questions or need further clarification, feel free to ask.

What Is a Derivative in Calculus? | Outlier (2024)
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