Find the derivative of each of the following functions with respect to $x$. \(f(x) = \ln \frac{\co

Question

Find the derivative of each of the following functions with respect to $x$.

$f(x) = \ln \frac{\cos (x^2 + x + 1)}{\sqrt{\sin (2x^2 + 1)}}$
$f(x) = \ln \frac{\tan (1 - 2x^2)}{\sqrt{1 - x}}$
$f(x) = \ln \sqrt{\frac{\sin (1 - x - x^2)}{\cos 3x - \sin 5x}}$
$f(x) = \ln \sqrt{\frac{\sin 2x}{\cos^2 x}}$

Accepted Answer

(i) $-(2x+1)\tan(x^2+x+1)-2x\cot(2x^2+1)$ (ii) $-4x\dfrac{\sec^2(1-2x^2)}{\tan(1-2x^2)}+\dfrac{1}{2(1-x)}$ (iii) $\dfrac12\left[-(1+2x)\cot(1-x-x^2)+\dfrac{3\sin 3x+5\cos 5x}{\cos 3x-\sin 5x}\right]$ (iv) $\cot 2x+\tan x$

Techniques of Differentiation - Derivatives of Natural Logarithmic Functions

Techniques of Differentiation - Derivatives of Natural Logarithmic Functions

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