Differentiation Techniques

Definition
Differentiation

The definition of the derivative for the function \(f(x)\) is based upon the limit of the average rate of change as shown below.

\(\displaystyle\underset{n\to\infty}{\lim}f'(x)=\frac{f(x+h)-f(x)}{h}\)

CAUTION: Questions that direct a student to compute a derive by using the limit definition are not permitted to apply another technique such as the well celebrated power rule.

Linearity Property
Differentiation

The differentiation operator \(\frac{d}{dx}\)is type operator known as a linear operator.

If \(f(x),g(x)\) are differential functions then the following holds

\(\displaystyle\big(f\pm g\big)'=f'\pm g'\)
\(\displaystyle\big(k f\big)'=k\big(f\big)'\)

Power Rule
Differentiation

The power rule is useful for computation the derivative of terms with power solely upon the variable, such as \(x^n\)

\(\displaystyle\frac{d}{dx}\big(x^n\big) = nx^{n-1}\)

Note \(\,\frac{d}{dx}\big(x\big)=1,\, \frac{d}{dx}\big(k\big)=0\)

Properties of exponents maybe needed to recalibrate a term such as \(\frac{1}{\sqrt{x}}=x^{\frac{-1}{2}}\)

Product Rule
Differentiation

The product rule is used whenever... WHENEVER... differentiating the product of two differentiable functions \(f(x),g(x)\)

\( \big(fg\big)' = f'g + fg'\)

Quotient Rule
Differentiation

Use the quotient rule when computing the derivative of the quotient of two differentiable functions \(f(x),g(x)\)

\(\displaystyle\left(\frac{f}{g}\right)' = \frac{f'g-fg'}{g^2} \)

Chain Rule
Differentiation

The chain rule is used for computing the derivative of a composition of differentiable functions \(f(x),g(x)\)

\(\big(f(g(x)) \big)' = f'\big(g(x)\big)\cdot g'(x)\)

Implicit Differentiation
Differentiation

Implicit differentiation is a special case of applying the chain rule to compute the derivative for a function/curve \(y\) that is not explicitly stated in terms of the variable of differentiation, \(x \).

To apply this technique, differentiate \(y\) as normal and 'multiply' by \(\frac{dy}{dx}\) whenever taking the derivative of \(y\) with respect to \(x\).

Logarthimc Differentiation
Differentiation

\(\underset{n\to\infty}{\lim}\frac{a_n}{b_n}=L\)

This technique is useful technique for computing derivatives by applying logarithmic properties to reduce the complexity of function such as exponentials, quotients, and products.

\(\texttt{ln}\big(AB\big)=\texttt{ln}\big(A\big)+\texttt{ln}\big(B\big)\)
\(\texttt{ln}\left(\frac{A}{B}\right)=\texttt{ln}\big(A\big)-\texttt{ln}\big(B\big)\)
\(\texttt{ln}\left(A^n\right)=n\texttt{ln}\left(A\right)\)

Parametric Equations
Differentiation

To compute the derivatives of a function (or curve) that has been parametrize as \(y:=y(t)\) and \(x:=x(t)\) can be found as follows

\(\displaystyle\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}} \)

\(\displaystyle\frac{d^2y}{dx^2}=\frac{ \frac{d}{dt}\big(\frac{dy}{dx} \big) }{\frac{dx}{dt}} \)

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