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.

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.

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:

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.

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.

The derivative of sin^3x (read as sin cube x) is equal to 3 sin²x 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(xⁿ)/dx = nx^n-1 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^3-1x × d(sinx)/dx

= 3 sin²x cosx

Hence, the derivative of sin cube x is equal to 3 sin²x 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 sin²x cosx.
• We can evaluate the sin3x differentiation using the chain rule and first principle of derivatives.

☛ Related Topics:

• Derivative of Cosx
• Derivative of 2x
• Derivative of Sin inverse x

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 sin²x 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^3-1x × d(sinx)/dx = 3 sin²x 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, d²(sin3x)/dx² = d(3 cos3x)/dx = -9 sin3x.