Page 287 - DMTH201_Basic Mathematics-1

Basic Mathematics – I

Notes Alternate 2
Using logarithmic differentiation we have:
ln y = ln (2x) sin x = sin x ln 2x,
y' 2 sin x
y cos x ln 2x (sin ) x 2x cos x ln 2x x ,
sin x sin x sin x
y' y cos x ln 2x 2 ( ) x cos x ln 2x .
x x

g(x)
Notes The given function is of the form ( f(x)) , with f(x) = 2x and g(x) = sin x. The variable
a–1
a
appears in both the base and the exponent. Neither the power rule (d/dx) u = au u’ nor
u
u
the exponent rule (d/dx) a = a ( ln a)u’ can be applied directly in this case.
g(x)
v
In Solution 1 we transform ( f(x)) into the exponential function using the definition u =
g(x)
e v ln u , to get ( f(x)) = e g(x) ln f(x) . Then we differentiate e g(x) ln f(x) with respect to x utilizing the
exponent rule. This is possible because the base e is a constant. In the answer, we transform
g(x)
e g(x) ln f(x) back to ( f(x)) .
g(x)
In Solution 2 we take the natural logarithm of both sides of the equation y = ( f(x)) , to
g(x)
obtain ln y = ln ( f(x)) = g(x) ln f(x). Then we differentiate implicitly both sides of the
resulting equation ln y =g(x) ln f(x) with respect to x. Note that (d/dx) ln y = (1/y) dy/dx =
g(x)
y’/y, by the chain rule. Next we solve for y’. In the answer, we replace y by ( f(x)) , since
we should express y’ in terms of x only, not of x and y. This technique is called logarithmic
differentiation, since it involves the taking of the natural logarithm and the differentiation
of the resulting logarithmic equation. It allows us to convert the differentiation of ( f(x)) g(x)
into the differentiation of a product.
Note that both Alternate 1 and 2 yield the same answer.
3
Example: Find dy/dx if y = x (sin x) cos x .
Solution:
Alternate 1
3 cos x ln sin x
3
y = x (sin x) cos x = x e ,
dy 2 cos x ln sin x 3 cos x ln sin x cos x
3 e x e ( sin ) x ln sin x cos x
x
dx sin x
3x 2 (sin ) x cos x x 3 (sin ) x cos x ( sin x ln sin x cos x cot ) x
3
cos
x
x 3 (sin ) x sin x ln sin x cos x cot x .
x
Alternate 2
Utilizing logarithmic differentiation we get:
3
ln y = ln x (sin x) cos x = 3 ln x + cos x ln sin x,
1 dy 3 ( sin ) x ln sin x cos x cos x 3 sin x ln sin x cos x cot , x
y dx x sin x x
dy 3 3 cos x 3
y sin x ln sin x cos x cot x x (sin ) x sin x ln sin x cos x cot x .
dx x x

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