Derivative of Cos x

The differentiation of cos x is the process of evaluating the derivative of cos x or determining the rate of change of cos x with respect to the variable x. The derivative of the cosine function is written as (cos x)' = -sin x, that is, the derivative of cos x is -sin x. In other words, the rate of change of cos x at a particular angle is given by -sin x.

Now, the derivative of cos x can be calculated using different methods. It can be derived using the limits definition, chain rule, and quotient rule. In this article, we will calculate the derivative of cos x and also discuss the anti-derivative of cos x which is nothing but the integral of cos x.

The derivative of cos x is the negative of the sine function, that is, -sin x. Derivatives of all trigonometric functions can be calculated using the derivative of cos x and derivative of sin x. The derivative of a function characterizes the rate of change of the function at some point. The process of finding the derivative is called differentiation. The differentiation of cos x can be done in different ways and it can be derived using the definition of the limit, and quotient rule. Since the derivative of cos x is -sin x, therefore the graph of the derivative of cos x will be the graph of the negative of -sin x.

Derivative of Cos x - Formula

Now, we will write the derivative of cos x mathematically. The derivative of a function is the slope of the tangent to the function at the point of contact. Hence, -sin x is the slope function of the tangent to the graph of cos x at the point of contact. Mostly, we memorize the derivative of cos x. An easy way to do that is knowing the fact that the derivative of cos x is negative of sin x and the derivative of sin x is the positive value of cos x. The expression to write the differentiation of cos x is:

d(cos x )/ dx = -sin x

As the derivative of cos x is negative of sin x, therefore graph of the derivative of cos x is similar to the graph of the trigonometric function sin x with negative values where sin x has positive values. First, let us see how the graphs of cos x and the derivative of cos x look like. As sin x is a periodic function, the graph of differentiation of cos x is also periodic and its period is 2π.

A derivative is simply a measure of the rate of change. Now, we will derive the derivative of cos x by the first principle of derivatives, that is, the definition of limits. To find the derivative of cos x, we take the limiting value as x approaches x + h. To simplify this, we set x = x + h, and we want to take the limiting value as h approaches 0. We are going to use certain trigonometry formulas to determine the derivative of cos x. The formulas are:

• cos (A + B) = cos A cos B - sin A sin B
• \(\lim_{x\rightarrow 0} \dfrac{\cos x -1}{x} = 0\)
• \(\lim_{x\rightarrow 0} \dfrac{\sin x}{x} = 1\)

Thus, we have

\(\begin{align}\frac{\mathrm{d} (\cos x)}{\mathrm{d} x} &= \lim_{h\rightarrow 0} \dfrac{\cos (x + h)-\cos x}{(x+h)-x} \\&= \lim_{h\rightarrow 0} \dfrac{\cos x \cos h -\sin x \sin h-\cos x}{h}\\&=\lim_{h\rightarrow 0} \dfrac{\cos h -1 }{h}\cos x - \dfrac{\sin h}{h}\sin x\\&=(0)\cos x - (1)\sin x\\&=-\sin x\end{align}\)

Hence the derivative of cos x has been proved using the first principle of differentiation.

The chain rule for differentiation is: (f(g(x)))’ = f’(g(x)) . g’(x). Now, to evaluate the derivative of cos x using the chain rule, we will use certain trigonometric properties and identities such as:

• \(\cos (\dfrac{\pi}{2} - \theta) = \sin \theta\)
• \(\sin (\dfrac{\pi}{2} - \theta) = \cos \theta\)
• d(sin x)/dx = cos x

Using the above three trigonometric properties, we can write the derivative of cos x as the derivative of sin (π/2 - x), that is, d(cos x)/dx = d (sin (π/2 - x))/dx . Using chain rule, we have,

\(\begin{align} \frac{\mathrm{d} \cos x}{\mathrm{d} x} &=\frac{\mathrm{d} \sin(\dfrac{\pi}{2}-x)}{\mathrm{d} x}\\&=\cos(\dfrac{\pi}{2}-x).(-1)\\&=-\cos(\dfrac{\pi}{2}-x)\\&= -\sin x\end{align}\)

Hence, we have derived the derivative of cos x as -sin x using chain rule.

The quotient rule for differentiation is: (f/g)’ = (f’g - fg’)/g². To derive the derivative of cos x, we will use the following formulas:

• cos x = 1/sec x
• sec x = 1/cos x
• d(sec x)/dx = sec x tan x
• tan x = sin x/ cos x

Using the above given trigonometric formulas, we can write the derivative of cos x and the derivative of 1/sec x, that is, d(cos x)/dx = d(1/sec x)/dx, and apply the quotient rule of differentiation.

\(\begin{align} \frac{\mathrm{d} \cos x}{\mathrm{d} x} &=\frac{\mathrm{d} (\dfrac{1}{\sec x})}{\mathrm{d} x}\\&=\dfrac{(1)' \sec x - (\sec x)' 1}{\sec^2x}\\&=\dfrac{0. \sec x - \sec x \tan x}{\sec^2x}\\&=\dfrac{- \sec x \tan x}{\sec^2x}\\&=\dfrac{-\tan x}{\sec x}\\&=\dfrac{\frac{-\sin x}{\cos x}}{\frac{1}{\cos x}}\\&=-\sin x\end{align}\)

Hence, we have derived the derivative of cos x using the quotient rule of differentiation.

The anti-derivative of cos x is nothing but the integral of cos x. As the name suggests, anti-derivative is the inverse process of differentiation. The derivative of cos x is -sin x and the derivative of sin x is cos x. So, the anti-derivative of cos x is sin x + C and the anti-derivative of sin x is -cos x + C, where C is constant of integration. Hence, we have obtained the anti-derivative of cos x as sin x + C.

\(\int \cos x = \sin x + C\)

Important Notes of Derivative of cos x

• The derivative of cos x is -sin x
• The anti-derivative of cos x is sin x + C
• Derivative of cos x can be derived using the definition of limit, chain rule and quotient rule.

Related Topics on Derivative of cos x

• Applications of Derivatives
• Chain Rule Formula
• Limits
• Cosine Formulas

What is the Derivative of Cos x in Calculus?

The derivative of cos x is the negative of the sine function, that is, -sin x. The derivative of a function is the slope of the tangent to the function at the point of contact. The derivative of cos x can be calculated using different methods.

What is the Derivative of cos x.sin x?

The derivative of cos x. sin x can be calculated using the product rule of differentiation. d(cos x. sin x)/dx = (cos x)' sin x + cos x (sin x)' = -sin x.sin x + cos x. cos x = cos²x - sin²x = cos 2x. Hence, the derivative of cos x.sin x = cos 2x

What is the Second Derivative of Cos x?

The first derivative of cos x is -sin x. The second derivative of cos x is obtained by differentiating the first derivative of cos x, that is, -sin x. The derivative of -sin x is -cos x. Hence, the second derivative of cos x is -cos x.

How to Find the Derivative of Cos x?

The derivative of cos x can be obtained by different methods such as the definition of the limit, chain rule of differentiation, and quotient rule of differentiation. To determine the derivative of cos x, we need to know certain trigonometry formulas and identities.