Three Ways to Write a Derivative
A derivative can be described as a function, as a rate of change using Leibniz notation, or as the result of applying a derivative operator. These are different notations for the same mathematical idea. Being able to move between them helps you read a formula accurately and evaluate a rate of change at a chosen input.
Suppose \(y=f(x)\). The notation \(f'(x)\) names the derivative function. The notation \(\frac{dy}{dx}\) also represents the derivative of \(y\) with respect to \(x\). The operator notation \(\frac{d}{dx}[f(x)]\) says to apply the differentiation operation with respect to \(x\) to the expression \(f(x)\).
The notation \(f'(x)\) depends on the function name \(f\). Leibniz notation displays the dependent variable \(y\) and the input variable \(x\). Operator notation emphasizes the instruction to differentiate. The notation changes, but the quantity being represented does not.
Derivative Functions and Values at Inputs
A derivative formula such as \(f'(x)=\cos x\) gives the derivative at every input in the domain. Substituting a particular value \(a\) gives one number, \(f'(a)\). It is important not to confuse the original function value \(f(a)\) with the derivative value \(f'(a)\): the former is the height of the function, and the latter is its instantaneous rate of change.
For example, if \(f(x)=\sin x\), then \(f(\pi/3)=\sqrt{3}/2\), while \(f'(\pi/3)=1/2\). The first is a function value; the second is a slope. Unless an angle measure is stated otherwise, trigonometric derivative formulas use radians.
Why the Notations Agree
The equivalence follows from the definition of a derivative. For a function \(f\), its derivative at \(a\), when the limit exists, is defined by the difference quotient:
Now write the same relationship using \(y=f(x)\). At \(x=a\), the value of \(y\) is \(f(a)\). At \(x=a+h\), the value is \(f(a+h)\). Thus the change in \(y\) is \(f(a+h)-f(a)\), and the change in \(x\) is \(h\). Leibniz notation names the resulting derivative \(\frac{dy}{dx}\). The operator \(\frac{d}{dx}\) names the operation that produces this derivative from \(f(x)\).
Each form therefore refers to the same limit, with a different emphasis. This does not mean that every derivative exists: the limit must exist for the derivative at the input to be defined.
Basic Derivative Formulas in Three Forms
The following formulas apply to the basic functions shown. The trigonometric formulas assume angles are measured in radians. The logarithm formulas require a positive input; the base-\(b\) logarithm also requires \(b>0\) and \(b\ne 1\).
| Function | Derivative function | Operator form |
|---|---|---|
| \(y=\sin x\) | \(f'(x)=\cos x\) | \(\frac{d}{dx}[\sin x]=\cos x\) |
| \(y=\cos x\) | \(f'(x)=-\sin x\) | \(\frac{d}{dx}[\cos x]=-\sin x\) |
| \(y=e^x\) | \(f'(x)=e^x\) | \(\frac{d}{dx}[e^x]=e^x\) |
| \(y=b^x\), \(b>0\) | \(f'(x)=b^x\ln b\) | \(\frac{d}{dx}[b^x]=b^x\ln b\) |
| \(y=\ln x\), \(x>0\) | \(f'(x)=1/x\) | \(\frac{d}{dx}[\ln x]=1/x\) |
| \(y=\log_b x\), \(x>0\), \(b>0,\ b\ne1\) | \(f'(x)=1/(x\ln b)\) | \(\frac{d}{dx}[\log_b x]=1/(x\ln b)\) |
Here \(\ln\) denotes the natural logarithm. Since \(e^x\) is the exponential function with base \(e\), its derivative is itself. For a different positive base \(b\), the factor \(\ln b\) is part of the derivative. When \(b=e\), \(\ln e=1\), so the general exponential formula agrees with the formula for \(e^x\).
Worked Examples
Worked Example: Sine in Three Notations
Let \(y=\sin x\). Write its derivative in three forms and evaluate it at \(x=\pi/6\).
First use the sine derivative formula, which assumes radians:
Thus, if \(f(x)=\sin x\), then \(f'(x)=\cos x\). Since \(y=\sin x\), Leibniz notation gives \(\frac{dy}{dx}=\cos x\). Now substitute \(x=\pi/6\):
Check: Each notation gives the same derivative value, \(\sqrt{3}/2\). The original function value is \(\sin(\pi/6)=1/2\), which is different, as expected because it is not the derivative value.
Worked Example: A Non-Natural Exponential
Let \(f(x)=2^x\). Find \(f'(x)\) and evaluate the derivative at \(x=3\), expressing the answer exactly.
For a positive base \(b\), the derivative is \(b^x\ln b\). Here \(b=2\), so
In operator notation, \(\frac{d}{dx}[2^x]=2^x\ln 2\). In Leibniz notation, if \(y=2^x\), then \(\frac{dy}{dx}=2^x\ln 2\). Substituting \(x=3\) gives
Check: The derivative formula evaluated at \(3\) is \(2^3\ln 2\), and \(2^3=8\), so all three notations give \(8\ln 2\). The factor \(\ln 2\) remains because the base is \(2\), not \(e\).
Worked Example: Natural Logarithm at a Positive Input
Let \(y=\ln x\). Find the derivative at \(x=e^2\), and write the result in all three notations.
The function \(\ln x\) is defined for \(x>0\), and \(e^2>0\), so the derivative formula applies. Since \(\frac{d}{dx}[\ln x]=1/x\),
Evaluate at \(x=e^2\):
Check: Substituting \(e^2\) into the derivative function \(1/x\) gives \(1/(e^2)\), so the evaluated result is \(e^{-2}\). The input is in the domain of \(\ln x\).
Worked Example: A Logarithm with Another Base
Let \(f(x)=\log_3 x\). Find the derivative at \(x=9\).
The base \(3\) is positive and not equal to \(1\), and the input \(9\) is positive. Therefore, the base-\(b\) logarithm formula applies:
In the other notations, \(\frac{d}{dx}[\log_3 x]=\frac{1}{x\ln 3}\), and if \(y=\log_3 x\), then \(\frac{dy}{dx}=\frac{1}{x\ln 3}\). At \(x=9\),
Check: Substituting \(x=9\) into \(1/(x\ln 3)\) gives exactly \(1/(9\ln 3)\). The input satisfies the logarithm's domain condition.
Common Notation Errors
A frequent error is to report \(f(a)\) when the question asks for \(f'(a)\). These quantities answer different questions. For \(f(x)=\cos x\) at \(x=0\), the function value is \(\cos 0=1\), while the derivative value is \(f'(0)=-\sin 0=0\).
Another common error is to omit the natural-logarithm factor when differentiating an exponential with a base other than \(e\). For example, the incorrect calculation \(\frac{d}{dx}[3^x]=3^x\) treats the base as though it were \(e\). The correction is
The check is to use the general formula \(b^x\ln b\) with \(b=3\). Similarly, differentiating \(\log_b x\) requires the factor \(\ln b\) in the denominator; the formula \(1/x\) is specifically for \(\ln x\).
Check Your Understanding
For each question, pay attention to the notation and to whether the answer should be a function or a number.
- If \(f(x)=\cos x\), write \(f'(x)\), \(\frac{dy}{dx}\) for \(y=\cos x\), and \(\frac{d}{dx}[\cos x]\).
- For \(g(x)=e^x\), evaluate \(g'(0)\) and express the value in operator notation at \(x=0\).
- If \(h(x)=5^x\), write its derivative function and evaluate it at \(x=1\).
- State the domain condition needed to use the derivative formula for \(\ln x\), and evaluate its derivative at \(x=1\).
- Identify the error in the claim \(\frac{d}{dx}[\log_2 x]=1/x\), then write the correct derivative formula.
