Power of x
$$\frac{d}{dx}\text{(constant)} = 0 \quad \frac{d}{dx} \left(x\right) = 1 $$
$$\frac{d}{dx} \left(x^n\right) = nx^{n-1} $$
Exponential
\begin{align*}
\frac{d}{dx}\left( e^x \right) &= e^x &&& \frac{d}{dx}\left( a^x \right) &= a^x \ln a \\ \\
\end{align*}
Trigonometric
\begin{align*}
\frac{d}{dx}\left(\sin x\right) &= \cos x &&& \frac{d}{dx}\left(\csc x\right) &= -\csc x \cot x \\ \\
\dfrac{d}{dx}\left(\cos x\right) &= -\sin x &&& \frac{d}{dx}\left(\sec x\right) &= \sec x \tan x \\ \\
\dfrac{d}{dx}\left(\tan x\right) &= \sec^2 x &&& \frac{d}{dx}\left(\cot x\right) &= -\csc^2 x
\end{align*}
Notice that a negative sign appears in the derivatives of the co-functions: cosine, cosecant, and cotangent.
Constant Factor Rule
Constants come out in front of the derivative, unaffected:
$$\dfrac{d}{dx}\left[c f(x) \right] = c \dfrac{d}{dx}f(x) $$
For example, $\dfrac{d}{dx}\left(4x^3\right) = 4 \dfrac{d}{dx}\left(x^3 \right) =\, … $
Sum of Functions Rule
The derivative of a sum is the sum of the derivatives:
$$\dfrac{d}{dx} \left[f(x) + g(x) \right] = \dfrac{d}{dx}f(x) + \dfrac{d}{dx}g(x) $$
For example, $\dfrac{d}{dx}\left(x^2 + \cos x \right) = \dfrac{d}{dx}\left( x^2\right) + \dfrac{d}{dx}(\cos x) = \, …$
Product Rule
\begin{align*}
\dfrac{d}{dx}(fg)&= \left(\dfrac{d}{dx}f \right)g + f\left(\dfrac{d}{dx}g \right)\\[8px]
&= {\small\Big[\text{ (derivative of the first) } \times \text{ (the second) }\Big] + \Big[\text{ (the first) } \times \text{ (derivative of the second)}\Big]}
\end{align*}
IV. Quotient Rule
\begin{align*}
\dfrac{d}{dx}\left(\dfrac{f}{g} \right) &= \dfrac{\left(\dfrac{d}{dx}f \right)g – f\left(\dfrac{d}{dx}g \right)}{g^2} \\[8px]
&={\small\dfrac{{\Big[\text{(derivative of the numerator) } \times \text{ (the denominator)}\Big] – \Big[\text{ (the numerator) } \times \text{ (derivative of the denominator)}}\Big]}{\text{all divided by [the denominator, squared]}}}
\end{align*}
Many students remember the quotient rule by thinking of the numerator as “hi,” the demoninator as “lo,” the derivative as “d,” and then singing
“lo d-hi minus hi d-lo over lo-lo”
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Jump down this page to: [Power rule: $x^n$] [Exponential: $e^x$] [Trig derivs] [Product rule] [Quotient rule] [More problems & University exam problems]
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