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.

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.

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:

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.

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.

The derivative of cos cube x is equal to -3 cos²x 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^3-1x × d(cosx)/dx

= 3 cos²x × (-sinx)

= -3 cos²x 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 cos²x sinx.
• We can evaluate the differentiation of cos3x and cos^3x using the first principle of derivatives and chain rule method.

☛ Related Topics:

• Differentiation and Integration
• Antiderivative Rules
• Sin2x Formula

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 cos²x 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.