Derivative of A Function With Respect To Another F
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Derivative of A Function With Respect To Another Function
The technique of differentiating a function defined parametrically is also applicable in finding the derivative of one function with respect to another function. For example, suppose we wish to find the derivative of f (x) with respect to g (x). Then by letting.
y = f (x), z = g (x)
and treating x as the parameter, we have
dv = dy/dx = f ‘(x)
dz dz/dx g '(x)
Example . Differentiate
(i) (logx)2 with respect to ex .
(ii) x2 + log x + e-x with respect to √x.
Solution. (i) Let y = (log x )2 and z = ex .
Then we have to find dy/dz. Now
dy = 2 (logx) 1 = 2 logx
dx x x
and dz/dx = ex
Thus dy = dy/dx = 2 log x)/ x = 2 log x
dz dz/dx ex xex
(iii) Let y = x2 + logx +e-x and z = √x
Then dy = 2x + 1 - e -x and dz = 1
dx x dx 2√x
Hence dv = dv/dx = 2√x . ( 2x + 1 -e -x ) .
dz dz/dx x
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y = f (x), z = g (x)
and treating x as the parameter, we have
dv = dy/dx = f ‘(x)
dz dz/dx g '(x)
Example . Differentiate
(i) (logx)2 with respect to ex .
(ii) x2 + log x + e-x with respect to √x.
Solution. (i) Let y = (log x )2 and z = ex .
Then we have to find dy/dz. Now
dy = 2 (logx) 1 = 2 logx
dx x x
and dz/dx = ex
Thus dy = dy/dx = 2 log x)/ x = 2 log x
dz dz/dx ex xex
(iii) Let y = x2 + logx +e-x and z = √x
Then dy = 2x + 1 - e -x and dz = 1
dx x dx 2√x
Hence dv = dv/dx = 2√x . ( 2x + 1 -e -x ) .
dz dz/dx x
For more help in Derivative of A Function With Respect To Another Function click the button below to submit your homework assignment