(1) Constant rule:

If the function f is a constant, then its derivative is zero. Moreover if a function is multiplied by a constant then its derivative is given by

{cf(x)}' = cf'(X)

(2) Sum Rule:

{f(x) ± g(x)}' = f'(x) ± g'(x)

(αf + βg)' = αf' + βg' , for all functions f and g and real numbers α and β.

(3) Product Rule:

(fg)' = f'g + g'f, for all functions f and g.

(4) Quotient Rule:

(f/g)' = (f'g – fg')/g2, for all functions f and g such that g ≠ 0.

(5) Chain Rule:

If f(x) = h(g(x)), then f'(x) = h'(g(x)). g'(x)

So while using the chain rule remember the following points:

(i) Express the original function as a simpler function of u, where u is a function of x.

(ii) Differentiate the two functions you now have. Multiply the derivatives together, leaving your answer in terms of the original function (i.e. in x's rather than u's).


Logarithmic Differentiation
To differentiate some special functions using logarithm is called Logarithm Differentiation. When it is difficult to differentiate the function then we use the differentiation using logarithms.
Logarithm Differentiation starts with taking the natural logarithm that is, logarithm to the base e on the both
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The second and third derivatives
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Example

Differentiate y = xx

Solution:

y = xx

We will take the logarithm to both sides

In y = In xx

= x In x

Now we will differentiate both sides by using chain rule to the left-hand side as y represents a function of x and use the product rule on the right-hand side.

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