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Derivative of Sin3x
The derivative of sin3x is equal to 3cos3x. We can evaluate the differentiation of sin3x using different methods of derivatives such as the first principle of derivatives and the chain rule method. We know that the derivative of sin(ax) is equal to a times cos(ax), that is, d(sinax)/dx = a cos(ax) which is determined using the chain rule of derivatives. Substituting a = 3 into this formula, we have the derivative of sin3x as d(sin3x)/dx = 3 cos3x.
Further in this article, we will evaluate the derivative of sin^3x using the chain rule method of differentiation. We will also determine the formula for the derivative of sin3x using the first principle and the formula for the derivative of sin cube x and solve some examples related to the concept for a better understanding of the concept.
What is Derivative of Sin3x?
The derivative of sin3x is equal to three times cos3x, that is, d(sin3x)/dx = 3 cos3x. Differentiation of sin3x is the process of finding its derivative which can be determined using various differentiation methods. We can find the derivative of sin3x using the first principle of derivatives, that is, the definition of limits and the chain rule method of differentiation. In the next section, let us explore the formula for the derivative of sin3x.
Derivative of Sin3x Formula
Now, the formula for the derivative of sin3x is given by, d(sin3x)/dx = 3 cos3x. We use the chain rule method to find the derivative of composite functions and sin3x is a composite function of f(x) = sinx and g(x) = 3x given by fog(x). The image given below shows the formula of sin3x differentiation:
Derivative of Sin3x By First Principle
Now that we know the derivative of sin3x, in this section, we will evaluate the sin3x differentiation using the first principle of derivatives. We will use different formulas of derivatives and limits of trigonometry to prove that the derivative of sin3x is equal to 3 cos3x. To find the derivative of sin3x, 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 the following formulas:
 sin A  sin B = 2 cos ½ (A + B) sin ½ (A  B)
 lim_{x→0} (sinx)/x = 1
 d(f(x))/dx = lim_{h→0} [f(x+h)  f(x)]/h
Using the above formulas, we have
d(sin3x)/dx = lim_{h→0} [sin3(x+h)  sin3x]/h
= lim_{h→0} [sin (3x + 3h)  sin3x]/h
= lim_{h→0} {2 cos [(3x + 3h + 3x)/2] sin [(3x + 3h  3x)/2]}/h  [Using Sin A  Sin B Formula]
= lim_{h→0} {2 cos [(6x + 3h)/2] sin (3h/2)}/h
= lim_{h→0} {2 cos [(6x + 3h)/2] × (3/2) lim_{h→0 }sin (3h/2)}/(3h/2)  [Multiplying and dividing the limit by 3/2]
= 2 cos (6x/2) × (3/2) × 1 [Using the formula lim_{x→0} (sinx)/x = 1]
= 3 cos3x
Hence, we have proved that the differentiation of sin3x is equal to 3 cos3x by the first principle of differentiation.
Derivative of Sin3x Using Chain Rule Method
We use the chain rule method to find the derivatives of the composite functions. We know that sin3x is the composition of the functions f(x) = sinx and g(x) = 3x and is written as f(g(x)) = f(3x) = sin3x. To find the derivative of sin3x using the chain rule method, we write it as the product of the derivative of sin3x with respect to 3x and the derivative of 3x with respect to x.
d(sin3x)/dx = d(sin3x)/d(3x) × d(3x)/dx
= cos3x × 3  [Because derivative of sinx with respect to x is equal to x]
= 3 cos3x
Hence, the derivative of sin3x is equal to cos3x using the chain rule method.
What is the Derivative of Sin^3x?
The derivative of sin^3x (read as sin cube x) is equal to 3 sin^{2}x cosx. To prove the formula for the derivative of sin cube x, we will use the method of the chain rule. We use the power rule of differentiation given by d(x^{n})/dx = nx^{n1} the formula for the derivative of sinx that is given by d(sinx)/dx = cosx. Using these formulas, we have
d(sin^3x)/dx = 3 sin^{31}x × d(sinx)/dx
= 3 sin^{2}x cosx
Hence, the derivative of sin cube x is equal to 3 sin^{2}x cosx.
Important Notes on Derivative of Sin3x
 The derivative of sin3x is equal to 3 cos3x.
 The derivative of sin^3x is equal to 3 sin^{2}x cosx.
 We can evaluate the sin3x differentiation using the chain rule and first principle of derivatives.
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Derivative of Sin3x Examples

Example 1: What is the derivative of sin^3x with respect to cos^3x?
Solution: To find the derivative of sin3x w.r.t. cos^3x, assume u = sin^{3}x and v = cos^{3}x. Now, we need to find the value of du/dv.
First, differentiate sin3x with respect to x, that is,
du/dx = d(sin^{3}x)/dx
= 3 sin^{2}x cosx [Because derivative of sin^3x is 3 sin^{2}x cosx] → (1)
Now, find the derivative of cos^3x with respect to x, that is,
dv/dx = d(cos^{3}x)/dx
= 3 cos^{2}x (sinx) [Because the derivative of cosx is sinx]
= 3 cos^{2}x sinx → (2)
Now, to find the derivative of sin^3x w.r.t. cos^3x, we have
du/dv = (du/dx)/(dv/dx)
= 3 sin^{2}x cosx/(3 cos^{2}x sinx) [From (1) and (2)]
=  sinx/cosx
=  tanx
Answer: The derivative of sin^3x with respect to cos^3x is equal to  tanx.

Example 2: Determine the second derivative of sin3x.
Solution: To find the second derivative of sin3x, we will differentiate its first derivative.
The first derivative of sin3x is equal to 3 cos3x. To find its second derivative, we have
d^{2}(sin3x)/dx^{2} = d(3 cos3x)/dx
= 3 d(cos3x)/dx
= 3 × d(cos3x)/d(3x) × d(3x)/dx
= 3 ×  sin3x × 3
= 9 sin3x
Answer: Hence, the second derivative of sin3x is equal to 9 sin3x.

Example 3: Find the derivative of sin3x cos3x.
Solution: To find the derivative of sin3x cos3x, we will use the sin2A formula of trigonometry. We know that sin2A = 2 sinA cosA which implies sinA cosA = (1/2) sin2A. Therefore, we can write sin3x cos3x as sin3x cos3x = (1/2) sin6x. Hence, we have
d(sin3x cos3x)/dx = d((1/2) sin6x)/dx
= (1/2) d(sin6x)/dx
= (1/2) 6 cos6x
= 3 cos6x
Answer: The derivative of sin3x cos3x is equal to 3 cos6x.
FAQs on Derivative of Sin3x
What is the Derivative of Sin3x in Calculus?
The derivative of sin3x is equal to 3 cos3x. We can evaluate this derivative using different methods of differentiation such as the chain rule method. Derivative of a function gives the rate of change in the function with respect to a small change in the variable.
What is the Formula for the Derivative of Sin3x?
The formula for the derivative of sin3x is given by, d(sin3x)/dx = 3 cos3x. As we know that the derivative of sin(ax) = a cos(ax) can be evaluated using the chain rule, therefore if we substitute a = 3 into this formula, we can get the derivative of sin3x.
How to Find the Derivative of Sin3x?
The derivative of sin3x can be determined using the first principle of derivatives and the chain rule method. We know that we use the chain rule for the derivatives of composite functions. Sin3x is the composition of two functions sinx and 3x.
What is Derivative of Sin^3x?
The derivative of sin^3x is equal to 3 sin^2x cosx whose formula can be written as d(sin^3x)/dx = 3 sin^{2}x cosx.
How to Find the Derivative of Sin Cube x?
We can calculate the derivative of sin cube x using the power rule of differentiation, the derivative of sinx formula, and the chain rule method. It is evaluated as d(sin^3x)/dx = 3 sin^{31}x × d(sinx)/dx = 3 sin^{2}x cosx.
What is the Second Derivative of Sin3x?
The second derivative of sin3x is equal to 9 sin3x. This can be determined by taking the derivative of the first derivative of sin3x, that is, d^{2}(sin3x)/dx^{2} = d(3 cos3x)/dx = 9 sin3x.
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