Find the derivative of cos^2(x) using the product rule
Find the derivative of cos^2(x) using the product rule

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Derivative of Cos Square x (Cos^2x)

The derivative of cos square x is equal to the negative of the trigonometric function sin2x. Mathematically, we can write this formula for the derivative of cos^2x as, d(cos2x) / dx = – sin2x (which is equal to -2 sin x cos x). The derivative of a function gives the rate of change of the function with respect to the variable. In other words, we can also say that the derivative of cos square x gives the function for the slope of the tangent to the function at the points of contact.

In this article, we will evaluate the derivative of cos square x and derive its formula using different methods of differentiation including the first principle of derivatives, chain rule, and product rule formula. We will also solve various examples involving the derivative of cos^2x for a better understanding of the concept.

What is Derivative of Cos Square x?

The derivative of cos square x is given by, d(cos2x) / dx = – sin2x. Generally, we can evaluate this derivative using the chain rule of differentiation (which will involve the use of the power rule and the derivative of cos x formula). The derivative of cos^2x gives the slope function of the tangent to the curve of cos2x. Other methods to evaluate the derivative of square x are the first principle of derivatives and using the product rule formula.

Derivative of Cos^2x Formula

The formula for the derivative of cos^2x is given by,

  • d(cos2x) / dx = -sin2x (OR)
  • d(cos2x) / dx = – 2 sin x cos x (because sin 2x = 2 sinx cosx).

We can evaluate these formulas using various methods of differentiation. Let us go through those derivations in the coming sections.

Derivative of Cos Square x Using the Chain Rule

According to the chain rule of differentiation, the derivative of the composition of functions is given by f(g(x)) = d [f(x)] / d [g(x)] × d [g(x)] / dx. Here, we have f(x) = x2 and g(x) = cos x. We will use the power rule and the derivative of cos x formula to apply the chain rule to find the derivative of cos square x. So, we have

d(cos2x) / dx = d(cos2x) / d(cos x) × d(cos x) / dx

= 2 cos x × -sin x — [Because dxn/dx = nxn-1 and derivative of cos x is – sinx]

= – 2 cos x sin x

= – sin 2x — [Because sin2x = 2 sin x cos x]

Hence, we have proved that the derivative of cos^2x is equal to – sin 2x

Derivative of Cos^2x Using the First Principle

In this section, we will derive the formula for the derivative of cos^2x using the first principle of differentiation (also known as the definition of limits). To find the derivative of cos square 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 will use some limit formulas and trigonometric formulas to simplify the problem:

  • f'(x) = limh→0 [f(x + h) – f(x)] / h
  • limh→0 (sin x) / x = 1
  • Algebraic identity: (a + b) (a – b) = a2 – b2
  • Cos A + Cos B = 2 cos ½ (A + B) cos ½ (A – B)
  • Cos A – Cos B = – 2 sin ½ (A + B) sin ½ (A – B)

Using the above formulas, we have

d(cos2x) / dx = limh→0 [cos2(x + h) – cos2(x)] / h

= limh→0 [(cos(x + h) + cos(x)) (cos(x + h) – cos(x))] / h

= limh→0 [(2 cos ½ (x + h + x) cos ½ (x + h – x)) (- 2 sin ½ (x + h + x) sin ½ (x + h – x))] / h

= – limh→0 [(2 cos ½ (2x + h) cos ½ (h)) (2 sin ½ (2x + h) sin ½ (h))] / h

= – limh→0 [(2 cos ½ (2x + h) sin ½ (2x + h)) (2 cos ½ (h) sin ½ (h))] / h

= – limh→0 [ sin (2x + h) × sin h] / h — [Using sin2x formula]

= – limh→0 sin (2x + h) × limh→0 (sin h) / h

= – sin 2x × 1

= – sin 2x

Hence, we have derived the formula for the derivative of cos square x using the first principle of differentiation, that is, the definition of limits.

Derivative of Cos Square x Using Product Rule

We can write cos square x as a product of cosine function with itself, that is, cos2x = cos x × cos x. The product rule formula of differentiation is given by, (uv)’ = u’v + uv’. Also, we know that the derivative of cos x is equal to – sinx and sin 2x = 2 cos x sin x. Using these facts and formulas, we have

(cos2x)’ = (cos x × cos x)’

= (cos x)’ cos x + cos x (cos x)’

= -sinx cosx + cosx (-sinx)

= -2 sin x cos x

= – sin2x

Therefore, we have proved that the derivative of cos^2x is equal to -sin2x.

Important Notes on Derivative of Cos Square x

  • The derivative of cos^2x is equal to -2 sin x cos x which is equal to – sin2x.
  • We can evaluate this derivative using the first principle of derivatives, product rule, and chain rule.

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Derivative of Cos Square x Examples

  1. Example 1: Find the derivative of cos^2x – sin2x.

    Solution: The derivative of cos2x – sin2x can be calculated using the chain rule formula.

    d(cos2x – sin2x) / dx = d(cos2x)/dx – d(sin2x)/dx

    = – sin2x – 2 sin x cos x — [because derivative of cos square x is -sin2x, derivative of sin x is cos x.]

    = -sin2x – sin2x

    = -2 sin2x

    Answer: d(cos2x – sin2x) / dx = -2 sin2x

  2. Example 2: Evaluate the derivative of cos square root x.

    Solution: To evaluate the derivative of cos square root x, we will apply the chain rule. We will use the following formulas:

    • Derivative of cos x = -sin x
    • Derivative of root x = (1/2√x)
    • Power rule: d(xn)/dx = nxn-1

    Using the above formulas, we have

    d[cos2(√x)] / dx = d(cos2(√x)) / d(cos(√x)) × d(cos(√x)) / d(√x) × d(√x) / dx

    = 2 cos (√x) × -sin (√x) × (1/2√x)

    = (-1/2√x) sin(2√x)

    Answer: d[cos2(√x)] / dx = (-1/2√x) sin(2√x)

  3. Example 3: What is the second derivative of cos square x?

    Solution: To find the second derivative of cos square x, we will differentiate the first derivative of cos^2x. So, we have

    d2(cos2x)/dx2 = d(-sin2x) / dx — [because the first derivative of cos^2x is -sin2x]

    = -2 cos2x

    Answer: d2(cos2x)/dx2 = -2 cos2x

Derivative of Cos^2x Questions

FAQs on Derivative of Cos Square x

What is the Derivative of Cos Square x in Calculus?

The derivative of cos square x is equal to the negative of the trigonometric function sin2x. The derivative of cos square x gives the function for the slope of the tangent to the function at the points of contact.

How to Find the Derivative of Cos^2x?

We can find the derivative of cos^2x using different methods of differentiation including the first principle of derivatives, chain rule, and product rule formula.

What is the Formula for the Derivative of Cos Square x?

Mathematically, we can write the formula for the derivative of cos^2x as, d(cos2x) / dx = – sin2x (which is equal to -2 sin x cos x). dx shows that the differentiation of cos^2x is with respect to the variable x.

What is the Derivative of Cos^2x + 1?

The derivative of cos^2x + 1 is given by, d(cos2x + 1)/dx = (cos2x)’ + (1)’ = -sin2x + 0 = -sin2x. So, the derivative of cos^2x + 1 is equal to -sin2x.

Find the Derivative of Cos Square x Cube.

The derivative of cos square x cube is given by, d(cos2(x3))/dx = 2 cos(x3) × -sin(x3) × 3×2 = -3 sin(2×3). So, the derivative of cos square x cube is equal to -3 sin(2×3).

How to Find the Second Derivative of Cos^2x?

We can evaluate the second derivative of cos square x by differentiating the first derivative of cos^2x. The derivative of cos^2x is -sin2x. By differentiating this with respect to x, we obtained the second derivative of cos square x as d2(cos2x)/dx2 = -2 cos2x.

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