Derivative of Cos3x
The derivative of cos3x is given by 3 sin3x. We can evaluate the differentiation of cos3x using various methods of differentiation. The derivative of a function gives the rate of change in the function for a small change in the variable of the function. We know that the derivative of cosx is equal to sinx. Therefore, using the chain rule of differentiation, we can calculate the derivative of cos3x by taking the product of the derivative of cos3x with respect to 3x and the derivative of 3x with respect to x which is mathematically written as d(cos3x)/dx = d(cos3x)/d(3x) × d(3x)/dx = 3 sin3x.
In this article, we will find the derivative of cos^3x using the method of the chain rule. We will prove that the derivative of cos3x is equal to 3 sin3x using the first principle of derivatives and the chain rule method along with some solved examples for a better understanding of the concept.
What is Derivative of Cos3x?
The derivative of cos3x is equal to 3 sin3x. The derivative of a function gives the rate of change in that function with respect to the change in the variable. We can calculate the derivative of cos3x using different methods of differentiation such as the chain rule and first principle of derivatives. We know that the derivative of cos(ax) is equal to a sin(ax) using the chain rule method. Substituting a = 3 in this formula, we can get the derivative of cos3x to be equal to 3 cos3x. In the next section, let us go through the formula of the derivative of cos3x.
Derivative of Cos3x Formula
The formula for the derivative of cos3x is given by, d(cos 3x)/dx = 3 sin 3x. We can compute the differentiation of cos3x using the fact that the derivative of composite function h(x) = cos(ax) is equal to a sin(ax) as cos(ax) is the composition of functions f(x) = cosx and g(x) = 3x. The image given below shows the formula for the derivative of cos3x:
Derivative of Cos3x Using the Chain Rule Method
The chain rule method of differentiation is used to find the derivatives of composite functions. We can calculate the derivative of cos3x using the chain rule method. To evaluate the cos3x differentiation, we take the product of the derivative of cos3x with respect to 3x and the derivative of 3x with respect to x as follows:
d(cos3x)/dx = d(cos3x)/d(3x) × d(3x)/dx
= sin3x × 3
= 3 sin3x
Hence, we have proved that the derivative of cos3x is equal to 3 sin3x.
Derivative of Cos3x Using First Principle
We can calculate the derivative of cos3x using the first principle of derivatives. To find the differentiation of cos3x, 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 certain trigonometry and limit formulas to determine the derivative of cos3x:
 d(f(x))/dx = lim_{h→0} [f(x+h)  f(x)]/h
 cos (a + b) = cos a cos b  sin a sin b
 lim_{x→0} (sinx)/ x = 1
 lim_{x→0} (cos x  1) / x = 0
Using the above formulas, we have
d(cos3x)/dx = lim_{h→0} [cos3(x+h)  cos3x]/h
= lim_{h→0} [cos(3x + 3h)  cos3x]/h
= lim_{h→0} [cos3x cos3h  sin3x sin3h  cos3x]/h
= cos3x lim_{h→0} (cos3h  1)/h × (sin3x) lim_{h→0} (sin3h)/h
= cos3x lim_{h→0} 3 (cos3h  1)/3h × (sin3x) lim_{h→0} 3 (sin3h)/3h  [Multiplying and dividing the limit by 3]
= 3 cos3x lim_{h→0} (cos3h  1)/3h × 3(sin3x) lim_{h→0} (sin3h)/3h
= 3 cos3x × 0  3 sin3x × 1 → [Using the formulas of limits of sin and cos]
= 3 sin3x
Hence, we have proved that the differentiation of cos3x is 3 sin3x.
What is Derivative of Cos^3x?
The derivative of cos cube x is equal to 3 cos^{2}x sinx. We can calculate the derivative of cos^3x using the chain rule method. We will also use the power rule of differentiation and the formula for the derivative of cosx. Using these trigonometric and differentiation formulas, we have:
d(cos^3x)/dx = 3 cos^{31}x × d(cosx)/dx
= 3 cos^{2}x × (sinx)
= 3 cos^{2}x sinx
Hence, we have derived the formula for the derivative of cos3x.
Important Notes on Derivative Cos3x
 The formula for the derivative of cos3x is equal to 3 sin3x.
 The differentiation of cos^3x is equal to 3 cos^{2}x sinx.
 We can evaluate the differentiation of cos3x and cos^3x using the first principle of derivatives and chain rule method.
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Derivative of Cos3x Examples

Example 1: Evaluate the antiderivative of cos3x.
Solution: The antiderivative of cos3x is nothing but the integral of cos3x. To determine the antiderivative of cos3x, we will use the formula ∫cos ax dx = (1/a) sin ax + C, where C is the integration constant.
Substituting a = 3 in the above formula, we have
∫cos3x dx = (1/3) sin3x + C
Answer: The antiderivative of cos3x is equal to (1/3) sin3x + C.

Example 2: Compute the derivative of cos3x with respect to sin3x.
Solution: To find the derivative of cos3x with respect to sin3x, assume p = cos3x and q = sin3x. We need to determine the value of dp/dq. First, find the value of dp/dx and dq/dx.
dp/dx = d(cos3x)/dx
= 3 sin3x  Using the formula of derivative of cos3x
dq/dx = d(sin3x)/dx
= 3 cos3x  Using the formula of derivative of sin3x
dp/dq = (dp/dx)/(dq/dx)
= 3 sin3x/3 cos3x
=  tan3x
Answer: The derivative of cos3x with respect to sin3x is equal to tan3x.

Example 3: Evaluate the second derivative of cos3x.
Solution: The second derivative of cos3x can be determined by differentiating the first derivative of cos3x. Therefore, we have
d^{2}(cos3x)/dx^{2} = d(3 sin3x)/dx
= 3 d(sin3x)/dx
= 3 × 3 cos3x
= 9 cos3x
Answer: The second derivative of cos3x is equal to 9 cos3x.
FAQs on Derivative of Cos3x
What is the Derivative of Cos3x in Calculus?
The derivative of cos3x is equal to 3 sin3x. We can evaluate the differentiation of cos3x using various methods of differentiation. Mathematically, it is written as d(cos3x)/dx = d(cos3x)/d(3x) × d(3x)/dx = 3 sin3x.
How Do You Find the Derivative of Cos3x?
We know that the derivative of cos(ax) is equal to a sin(ax) using the chain rule method. Substituting a = 3 in this formula, we can get the derivative of cos3x to be equal to 3 cos3x. We can prove that the derivative of cos3x is equal to 3 sin3x using the first principle of derivatives and the chain rule method
What is the Second Derivative of Cos3x?
The second derivative of cos3x is equal to 9 cos3x. The second derivative of cos3x can be determined by differentiating the first derivative of cos3x.
What is the Derivative of Cos Cube x?
The derivative of cos^3x is equal to 3 cos^{2}x sinx. We can calculate the derivative of cos^3x using the chain rule method, the power rule of differentiation, and the formula for the derivative of cosx.
What is the Derivative of Cos3x w.r.t. Sin3x?
The derivative of cos3x with respect to sin3x is equal to tan3x.
What is the Antiderivative of Cos3x?
The antiderivative of cos3x is equal to (1/3) sin3x + C where C is the integration constant. The antiderivative of a function is nothing but its integral.
What is the Formula for the Derivative of Cos3x?
The formula for the derivative of cos3x is given by, d(cos 3x)/dx = 3 sin 3x. We can determine the cos3x differentiation using the fact that the derivative of composite function h(x) = cos(ax) is equal to a sin(ax) as cos(ax) is the composition of functions f(x) = cosx and g(x) = 3x.
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