# what is the derivative of cosx

Derivation of cos x Cos x differentiation 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, i.e. 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 by different methods. It can be inferred using the limit definition, series rule, and quotient rule. In this lesson, we will calculate the derivative of cos x and also discuss the derivative of cos x which is nothing but the integral of cos x. Read: What is the derivative of cosx 1. What is the derivative of cos x? 2. Graph of Derivative of Cos x 3. Derivation of Cos x Using first rule 4. Derivation of Cos x using series rule 5. Derivative of Cos x using quotient rule 6. Anti derivative of Cos x 7. Frequently asked questions about Derivative of Cos x The derivative of cos x is the negative of the sine function, ie -sin x. The derivative of all trigonometric functions can be calculated using the derivative of cos x and the derivative of sin x. The derivative of a function characterizes the rate of change of the function at some point in time. The process of finding the derivative is called discriminant. The distinction of cos x can be made in many different ways and it can be inferred using the definition of the limit and the quotient rule. Since the derivative of cos x is -sin x, then 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. Therefore, -sin x is the angular 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 to know for a fact that the derivative of cos x is negative of sin x and the derivative of sin x is a positive value of cos x. The differential expression for cos x is: d (cos x) / dx = -sin x Since the derivative of cos x is negative of sin x, the graph of the derivative of cos x is similar to that of the function. trigonometric sin x with a negative value where sin x has a positive value. First, let’s see what the graphs of cos x and the derivative of cos x look like. Since sin x is a periodic function, the discriminant graph of cos x is also periodic and its period is 2π.Read more: Verizon launches ‘FreeBee’ sponsored data program | The leading Q&AA derivative is simply a measure of the rate of change. Now, we’re going to take the derivative of cos x by the first principle of the derivative, which is the limit definition. To find the derivative of cos x, we take the limit value as x approaches x + h. To simplify this, we set x = x + h, and we want to take the limit value when h is close to zero. We will use some trigonometric formulas to determine the derivative of cos x. . The formulas are:

- cos(A + B) = cos A cos B – sin A sin B
- (lim_ {xrightarrow 0} dfrac {cos x -1} {x} = 0)
- (lim_ {xrightarrow 0} dfrac {sin x} {x} = 1)

Thus, we have (begin {align} frac {mathrm {d} (cos x)} {mathrm {d} x} & = lim_ {hrightarrow 0} dfrac {cos (x + h) -cos x} {( x + h) -x} &= lim_ {hrightarrow 0} dfrac {cos x cos h -sin x sin h-cos x} {h} &= lim_ {hrightarrow 0} dfrac {cos h -1} {h } cos x – dfrac {sin h} {h} sin x & = (0) cos x – (1) sin x &= – sin xend {align}) Hence the derivative of cos x has been proved by first principle usage of Difference The string rule for difference is: (f(g(x))) ‘=f'(g(x)). g'(x). Now, to evaluate the derivative of cos x using the series rule, we will use some trigonometric identity and properties like:

- (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), i.e. d(cos x) / dx = d(sin (π / 2). – x)) / dx. Using the sequence 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 xend {align }) We thus deduced the derivative of cos x to be -sin x using the sequence rule. Read more: What cheese does Chipotle use. The quotient rule to distinguish is: (f / g) ‘= (f’g – fg’) / g2. To calculate the derivative of cos x, we will use the following formulas:

- cos x = 1 / second x
- seconds x = 1 / cos x
- d (seconds x) / dx = seconds x tan x
- tan x = sin x / cos x

Using the trigonometric formulas given above, we can write the derivative of cos x and the derivative of 1/second x, that is, d(cos x) / dx = d(1/second x)/ dx, and apply the discriminant sequence rule. (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 xend {align}) Hence, we have deduced the derivative of cos x using the discriminant quotient rule. The antiderivative of cos x is nothing but the integral of cos x. As the name suggests, anti-derivative is the inverse differentiation process. 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 the integral constant. Thus, we have obtained that the opposite derivative of cos x is sin x + C. (int cos x = sin x + C)**Important Notes on Derivatives of cos x**

- The derivative of cos x is -sin x
- The opposite derivative of cos x is sin x + C
- The derivative of cos x can be deduced using the definition of the limit, the chain rule, and the quotient rule.

**Related topics about Derivatives of cos x**Read more: What is the voiceover of weed

- Applications of derivatives
- Chain rule formula
- Level
- Cosine Formula

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