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Derivative Formula: An Easy-To-Understand Guide

06.20.2022 • 7 min read

Rachel McLean

Subject Matter Expert

In this article, we’ll learn derivative formulas. We’ll discuss the limit definition of the derivative and introduce the most common derivative formulas. Finally, we’ll walk through examples of how to find the derivative of a function.

In This Article

  1. What Is a Derivative Formula?

  2. What Is a Derivative?

  3. Elements of a Derivative

  4. Key Derivative Formulas

  5. Solving Derivatives Step by Step

  6. Types of Derivatives

What Is a Derivative Formula?

Derivative formulas are equations that give quick solutions to common derivative problems. We refer to them as rules—like the power rule and the chain rule, to name a few.

More on these later.

These formulas come from the limit definition of the derivative, and they streamline the differentiation process. That’s why we can also call them differentiation formulas.

What Is a Derivative?

The derivative of a function at a point xx is equal to the slope of the tangent line at xx.

This slope value represents the instantaneous rate of change at that point. Differentiation is the process of determining the derivative of a function.

For example, in the graph below, the function f(x)=ln(x)f(x) = \ln{(x)} is in blue. The red line is f(x)=x1f(x) = x-1, which is the line tangent to ff at x=1x = 1. A tangent line to a point on a function is a line that just barely touches the function at that point. The slope of this tangent line f(x)=x1f(x) = x-1 is 11, which means that the derivative of f(x)=ln(x)f(x) = \ln{(x)} is 11 at x=1x = 1.

graph showing how to find Derivatives

We formally define derivatives using limits:

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} }=L

The above equation represents the limit of the average rate of change of ff over the interval [x,x+Δx][x, x +\Delta{x}] as Δx\Delta{x} approaches 0. We also know the average rate of change of a function as the slope of the secant line.

In this notation, Δx\Delta{x} represents a small change in xx. If this limit exists, then LL is the derivative.

Elements of a Derivative

The notation f(a)f’(a) represents the derivative of a function ff at some point aa. You might hear this notation read aloud as either “the derivative of ff evaluated at aa” or “ff prime at aa.”

The expressions f(x)f’(x) and dydx\frac{dy}{dx} both represent the general derivative function of ff. The latter notation is called Leibniz’s notation. By plugging any point aa into the resulting function f(x)f’(x), we can determine the slope of the tangent line of ff at any point on the curve.

Key Derivative Formulas

It’s essential to know how to use the limit definition to calculate a derivative. However, the limit definition can be clunky to use. Usually, we rely upon the standard derivative formulas below to differentiate, instead of using the formal limit definition:

Constant Rule

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

Power Rule

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

Special Case of the Power Rule

This case is where nn=1

ddx(x)=1\frac d{dx}(x)=1

Constant Multiple Rule

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

Chain Rule

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

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}

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)

Trigonometry Rules

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

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

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

Logarithmic Functions & Exponential Functions Rules

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

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

Two game-changing derivative rules, according to Dr. Tim Chartier, are the product rule and quotient rule:

He also explains more differentiation formulas for finding with examples:

Solving Derivatives Step By Step

Now, we’ll discuss how to find derivatives. We can solve a derivative in two ways:

  1. Using the formal limit derivative definition

  2. Using the basic derivative formulas

Here are the steps to finding derivatives using the limit definition:

Step 1

Substitute your function into the limit definition of a derivative 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} }=L

The hardest part of this step is the correct substitution of x+hx+h in the first term. You need to substitute xx with the expression (x+Δx)(x + \Delta{x}) wherever xx appears in f(x)f(x).

Step 2


Step 3

Evaluate the resulting limit.

Practice Example 1

For example, let’s find the derivative of the function f(x)=3x2f(x) = 3x^2.

Step 1 - Substitute

Let's sub in our function f(x)=3x2f(x) = 3x^2 into the limit definition of a derivative.

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} }
=limΔx03(x+Δx)23x2Δx= \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{3(x + \Delta{x})^2 - 3x^2}{\Delta{x}}

Step 2 - Simplify

We can do this by expanding the term 3(x+Δx)23(x + \Delta{x})^2 and then combining like terms. Then we can divide by Δx\Delta{x} since Δx\Delta{x} is present in all terms of the numerator and denominator.

f(x)=limΔx03(x2+2xΔx+Δx2)3x2Δxf’(x)= \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{3(x^2 + 2x\Delta{x} + \Delta{x}^2) - 3x^2}{\Delta{x}}
=limΔx03x2+6xΔx+3Δx23x2Δx= \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{3x^2 + 6x\Delta{x} + 3\Delta{x}^2 - 3x^2}{\Delta{x}}
=limΔx06xΔx+3Δx2Δx= \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{6x\Delta{x} + 3\Delta{x}^2}{\Delta{x}}
=limΔx06x+3Δx= \mathop {\lim }\limits_{\Delta{x} \to 0}6x + 3\Delta{x}

Step 3 - Evaluate

Now examine the limit as Δx\Delta{x} approaches 0. Polynomials are always continuous. To evaluate this limit, we can substitute Δx=0\Delta{x} = 0 directly into the function we’re left with.

f(x)=limΔx06x+3Δxf’(x)= \mathop {\lim }\limits_{\Delta{x} \to 0}6x + 3\Delta{x}
=6x+3(0)= 6x + 3(0)
=6x= 6x

So, we’ve found that the derivative of f(x)=3x2f(x) = 3x^2 is f(x)=6xf’(x) = 6x. This is the general derivative formula for any point on the curve of ff.

To find the derivative at a single point, we can plug x=ax = a into f(x)=6xf’(x) = 6x.

For example, we can say that:

  • f(1)=6(1)=6f’(1) = 6(1) = 6, which represents the slope of the tangent line at x=1x = 1.

  • f(5)=6(5)=30f’(5) = 6(5) = 30, which represents the slope of the tangent line at x=5x = 5.

  • f(60)=6(60)=360f’(-60) = 6(-60) = -360, which represents the slope of the tangent line at x=60x = -60.

  • f(0)=6(0)=0f’(0) = 6(0) = 0, which represents the slope of the tangent line at x=0x = 0.

Practice Example 2

We can also use the differentiation formulas to evaluate derivatives. For example, let’s find the derivative of f(x)=4x4+exf(x) = 4x^4 + e^x.

We can use the sum rule, which states that the derivative of a sum of functions is equal to the sum of their derivatives. We’ll also need to use the power rule for the 4x44x^4 term, where our exponent is n=4n = 4. Finally, we’ll use the exponential function rule for exe^x, which tells us that ddxex=ex\frac{d}{dx}e^x = e^x.

This gives us:

f(x)=ddx4x4+ddxexf’(x) = \frac{d}{dx}4x^4 + \frac{d}{dx}e^x
=44x41+ex=4 \cdot 4x^{4-1} + e^x
=16x3+ex= 16x^3 + e^x

Practice Example 3

For a slightly trickier example, let’s find the derivative of f(x)=sin(4x)cos(2x)f(x) = \sin{(4x)}\cos{(2x)}.

First, we’ll need the product rule. This says that the derivative of a product of functions is the sum of the first function times the derivative of the second, and the second function times the derivative of the first.

f(x)=sin(4x)ddxcos(2x)+cos(2x)ddxsin(4x)f’(x) = \sin{(4x)} \cdot \frac{d}{dx}\cos{(2x)} + \cos{(2x)} \cdot \frac{d}{dx}\sin{(4x)}

Along with the sine and cosine derivative rules, we’ll also need the chain rule, since we have a composition of functions using the sine and cosine functions.

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.

This means that the derivative of sin(4x)\sin{(4x)} is 4cos(4x)4\cos{(4x)}, and the derivative of cos(2x)\cos{(2x)} is 2sin(2x)-2\sin{(2x)}.

f(x)=sin(4x)ddxcos(2x)+cos(2x)ddxsin(4x)f’(x) = \sin{(4x)} \cdot \frac{d}{dx}\cos{(2x)} + \cos{(2x)} \cdot \frac{d}{dx}\sin{(4x)}
=sin(4x)2sin(2x)+cos(2x)4cos(4x)= \sin{(4x)} \cdot -2\sin{(2x)} + \cos{(2x)} \cdot 4\cos{(4x)}
=2sin(4x)sin(2x)+4cos(2x)cos(4x)= -2\sin{(4x)}\sin{(2x)} + 4\cos{(2x)}\cos{(4x)}
=4cos(2x)cos(4x)2sin(4x)sin(2x)= 4\cos{(2x)}\cos{(4x)} - 2\sin{(4x)}\sin{(2x)}

Types of Derivatives

First and second derivatives provide different information about the behavior of a function.

We use the sign of the first derivative to determine if a function is increasing, decreasing, or constant on an interval II:

  • If f(x)>0f’(x) > 0 for each xx on II, then ff is increasing on II.

  • If f’(x) < 0 for each x on I, then f is decreasing on I.

  • If f(x)=0f’(x) = 0 for each xx on II, then ff is constant on II.

1 graph with examples of concave up, concave down, and no concavity intervals

We find second derivatives by simply taking the derivative of the first derivative. Second derivatives inform us of the shape of a function. This characteristic is called concavity.

We use the sign of the second derivative to determine intervals of concavity:

  • If f’’(x)>0f’’(x) > 0 for each xx on II, then ff is concave up on II.

  • If f’’(x) < 0 for each x on I, then f is concave down on II.

  • If f’’(x)=0f’’(x) = 0 for each xx on II, then ff has no concavity.

Graph showing 4 different types of curve behavior with a function is increasing or decreasing and being concave up or down

Dr. Hannah Fry discusses first and second derivative tests:

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