Basic Derivative Rules

Basic Derivative Rules

The first-principles definition can become cumbersome when functions get complicated. Luckily, there are rules for finding derivatives more quickly depending on their form. These rules are derived from the first principles.

INTERACTIVE LAB

Compare a function with its derivative

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Function family powerconstantlogexpsincostancotseccscExponent

3

Input x

1

$\displaystyle f(1)=1,\qquad f'(1)=3$

Trigonometric inputs are in radians. Values outside the graph window remain available in the numerical readout.

Power Rule

THEOREM

Power Rule

If $f(x)=x^n$, then $f'(x)=nx^{n-1}$ wherever the real power is differentiable. For arbitrary real $n$, the rule applies on $x>0$; integer powers also apply elsewhere in their domains.

HANDS-ON PRACTICE

Differentiate a power

Easy·5 minShow reference answer

Find the derivative of $\displaystyle f(x) = x^7$

Derivative of a Constant

THEOREM

Derivative of a Constant

If $f(x)=k$, then $f'(x)=0$ Graphically, a constant function is a horizontal line, so its slope is zero.

HANDS-ON PRACTICE

Differentiate a constant

Easy·5 minShow reference answer

Find the derivative of $\displaystyle f(x) = 672$

Derivatives of Logarithmic Functions

THEOREM

Derivatives of Logarithmic Functions

For $x>0$, $a>0$, and $a\ne1$,

$$\begin{aligned} \frac{d}{dx}\log_{\scriptscriptstyle a} x&=\frac1{x\ln a}\\[.65em] \frac{d}{dx}\ln x&=\frac{d}{dx}\log_{\scriptscriptstyle e} x=\frac1{x\ln e}=\frac1x \end{aligned}$$

HANDS-ON PRACTICE

Differentiate a logarithm

Easy·5 minShow reference answer

Find the derivative of $\displaystyle f(x) = \log_{\scriptscriptstyle 4}(x)$

Derivatives of Exponential Functions

THEOREM

Derivatives of Exponential Functions

For $a>0$,

$$\begin{aligned} \frac{d}{dx}a^x&=a^x\ln a\\[.65em] \frac{d}{dx}e^x&=e^x\ln e=e^x \end{aligned}$$

The case $a=1$ is also valid: its derivative is zero.

Derivatives of Trigonometric Functions

THEOREM

Derivatives of Trigonometric Functions

Use radians, and restrict each formula to the function's domain.

$$\begin{aligned} \frac{d}{dx}\sin x&=\cos x\\[.65em] \frac{d}{dx}\csc x&=-\csc x\cot x\\[.65em] \frac{d}{dx}\cos x&=-\sin x\\[.65em] \frac{d}{dx}\sec x&=\sec x\tan x\\[.65em] \frac{d}{dx}\tan x&=\sec^2x\\[.65em] \frac{d}{dx}\cot x&=-\csc^2x \end{aligned}$$

Memory tool: if a trigonometric function starts with “c”, its derivative has a minus sign.

KNOWLEDGE QUIZ

Retry

QUESTION

What is $\displaystyle \dfrac{d}{dx}\log_{\scriptscriptstyle 4}x$ for $\displaystyle x>0$?

A$\displaystyle \dfrac1{x\ln4}$B$\displaystyle \dfrac1x$C$\displaystyle \dfrac{\ln4}x$D$\displaystyle 4^x\ln4$

 

HANDS-ON PRACTICE

Differentiate an exponential

Easy·5 minShow reference answer

Find the derivative of $\displaystyle f(x) = 5^x$