Question

Solve the following:
$\dfrac{d}{dx}\sqrt{{{\sec }^{2}}x+\cos e{{c}^{2}}x}=$

& (1)4\cos ec2x\cot 2x \\\ & (2)-4\cos ec2x\cot 2x \\\ & (3)4\cos ecx\cot 2x \\\ \end{aligned}$$ $$(4)$$None of these

Answer 1

By using the basic trigonometric identities firstly convert the given question into the simple form so that the differentiation of the function becomes easy to solve. Use the reciprocal identity of the trigonometric functions to convert into the basic identity of the trigonometry. Now you can solve the problem.

Complete step by step answer:
In this question we have to differentiate the function. But firstly we should know what the differentiation is, i.e. what is the meaning of the differentiation. Differentiation in mathematics simply means the rate of change of a function with respect to some variable. In the question we have to differentiate the function with respect to the variable$x$.
To differentiate any function there are certain rules. Firstly we need to understand that rule to differentiate the given functions. There are seven important rules and they are,
$1.$ Constant Function Derivative: If the given function is constant then its derivative is zero. For example, $f(x)=c$ where $c$is a constant then the derivative is,
$f'(x)=0$
$2.$ Power Function Derivative: If the given function has a constant power of any variable then its derivative is given by,
$f(x)={{x}^{c}}$where$c$is a constant
$f'(x)=c{{x}^{c-1}}$
$3.$Derivative of a function multiplied by a constant: If the given function is multiplied by some constant then its derivative is given by,
$f(x)=cx$ where $c$is a constant
$f'(x)=c\dfrac{d}{dx}x$
$4.$ Sum or difference derivative: If the given function is the sum or difference of two individual functions then its derivative is given by,
$f(x)=u(x)+v(x)$
$f'(x)=u'(x)+v'(x)$
$5.$ Chain rule: If the given function is the composite function then its derivative is given by,
$h(x)=f(g(x))$
$h'(x)=f'(g(x))g'(x)$
$6.$Product Rule: If the given function is the multiple of two individual functions then its derivative is given by,
$f(x)=u(x)v(x)$
$f'(x)=u'(x)v(x)+u(x)v'(x)$
$7.$ Quotient Rule: If the given function is the division of two individual functions then its derivative is given by,
$f(x)=\dfrac{u(x)}{v(x)}$
$f'(x)=\dfrac{u'(x)v(x)-u(x)v'(x)}{{{(v(x))}^{2}}}$
Using these rules of differentiation let us solve the question,
$\dfrac{d}{dx}\sqrt{{{\sec }^{2}}x+\cos e{{c}^{2}}x}$
Let the function be$f(x)=\sqrt{{{\sec }^{2}}x+\cos e{{c}^{2}}x}$, convert the form of the function using trigonometric identities for the ease of the calculation.
Using reciprocal identity the function$f(x)$becomes,
$f(x)=\sqrt{\dfrac{1}{{{\cos }^{2}}x}+\dfrac{1}{{{\sin }^{2}}x}}$
By taking the L.C.M., we get
$f(x)=\sqrt{\dfrac{{{\sin }^{2}}x+{{\cos }^{2}}x}{{{\sin }^{2}}x{{\cos }^{2}}x}}$
We know that,

& {{\sin }^{2}}x+{{\cos }^{2}}x=1 \\\ & \sin 2x=2\sin x\cos x \\\ \end{aligned}$$ Using these identities, we get $$\begin{aligned} & f(x)=\dfrac{1}{\dfrac{\sin 2x}{2}} \\\ & f(x)=\dfrac{2}{\sin 2x} \\\ \end{aligned}$$ Using reciprocal identity, we get $$f(x)=2\cos ec2x$$ Now the function gets transformed. Differentiate the function by using the differentiation rules and we get, $$\dfrac{d}{dx}f(x)=\dfrac{d}{dx}2\cos ec2x$$ $$\begin{aligned} & f'(x)=2\times (-\cos ec2x\cot 2x)\times 2 \\\ & f'(x)=-4\cos ec2x\cot 2x \\\ \end{aligned}$$ **So, the correct answer is “Option 2”.** **Note:** If the given function is of two variables then we will use the concept called partial differentiation. In this type of problem firstly we differentiate the function by considering one of the variables as constant and then we will differentiate the function by considering other variables as constant.