Question

If \[y = {\cos ^2}x\], then find\[\dfrac{{dy}}{{dx}}\].

Answer 1

Hint:
Here we will find the derivative of the given function with respect to \[x\] using the differentiation formula. We will further simplify it by using a trigonometric equation to get the required value. Differentiation is used to calculate the instantaneous rate of change in the function given because of one of its variables.

Complete step by step solution:
We have to find the derivation of \[y = {\cos ^2}x\] with respect to \[x\].
Now, starting with the differentiation of the cosine term we can write the equation as:
\[\dfrac{{dy}}{{dx}} = \dfrac{{d\left( {{{\cos }^2}x} \right)}}{{dx}}\]
On differentiating the function by using formula \[\dfrac{{d\left( {{x^n}} \right)}}{{dx}} = n{x^{n - 1}}\dfrac{{d\left( x \right)}}{{dx}}\] where \[x = {\cos ^2}x\], we get
\[ \Rightarrow \dfrac{{dy}}{{dx}} = 2\cos x\dfrac{{d\left( {\cos x} \right)}}{{dx}}\]
As we know differentiation of \[\cos x\] is \[ - \sin x\], so we get
\[\begin{array}{l} \Rightarrow \dfrac{{dy}}{{dx}} = 2\cos x \times \left( { - \sin x} \right)\\ \Rightarrow \dfrac{{dy}}{{dx}} = - 2\cos x\sin x\end{array}\]
Now we know that the relation between sine and cosine terms is \[\sin 2x = 2\sin x\cos x\].
Now, using this relation, we get
\[ \Rightarrow \dfrac{{dy}}{{dx}} = - \sin 2x\]

Additional information:
Trigonometry is a branch of mathematics that helps us to study the relationship between the sides and the angles of a triangle. In practical life, trigonometry is used by cartographers (to make maps). It is also used by the aviation and naval industries. In fact, trigonometry is even used by Astronomers to find the distance between two stars. Hence, it has an important role to play in everyday life. The three most common trigonometric functions are the tangent function, the sine and the cosine function. In simple terms, they are written as ‘sin’, ‘cos’ and ‘tan’.

Note:
Differentiation is done with respect to an independent variable of the function. Some real life applications of differentiation is rate of change of velocity with respect to time. It is also used to find the tangent and normal curve as also to calculate the highest and lowest point of the curve in a graph. Differentiation of a constant value is always zero because differentiation measures the rate of change of a function with respect to the variable but the constants don’t change their derivative so their differentiation is zero.