Introduction to the Derivative of 3sin x
Derivatives have a wide range of applications in almost every field of engineering and science. The derivative of 3sin x can be calculated by following the rules of differentiation. Or, we can directly find the 3sinx derivative by applying the first principle of differentiation. In this article, you will learn what the derivative of 3sin x is and how to calculate the differential of 3sinx formula by using different approaches.
What is the derivative of 3sinx?
The derivative of 3sin x with respect to the variable x can be calculated using the formula d/dx (3sin x) = 3cos x. This formula represents the rate of change of the trigonometric function sin x, which is widely used in geometry and mathematics. In fact, sin x represents the ratio of the opposite side of a triangle to its hypotenuse, making it a fundamental concept in trigonometry. By understanding how to differentiate 3sin x, you can better understand how changes in the variable x affect the value of the function.
Derivative of 3 sin x formula
The formula for the 3sin x differentiation is;
d/dx(3sinx) = 3cosx.
This means that the rate of change of the function 3sinx with respect to the variable x is equal to the cosine function of x multiplied by 3. By knowing this formula, you can easily calculate the derivative of 3sinx, which is useful in solving many calculus problems.
Related: Find more derivative problems and learn more.
How do you prove the derivative of 3 sinx?
There are various ways to prove the differentiation of 3sin x. These are;
First Principle
Chain Rule
Quotient Rule
Each method provids a different way to compute the 3sin x differentiation. By using these derivative rules, we can mathematically prove the formula for finding differential of 3sinx.
Derivative of 3sin x by first principle
According to the first principle of derivative, the 3 sin x derivative is equal to 3cos x. The derivative of a function by first principle refers to finding a general expression for the slope of a curve by using algebra. It is also known as the delta method. The derivative is a measure of the instantaneous rate of change, which is equal to,
f(x) = lim f(x+h)-f(x) / h
This formula allow us to determine the rate of change of a function at a specific point by using limit definition of derivative.
Proof of derivative of 3 sin x by first principle
To prove the derivative of 3sin(x) by using first principle, replace f(x) by 3sin x.
f(x) = limh→0f(x+h) - f(x) / h
f(x) = lim 3sin (x+h) - 3sin x / h
Moreover, you can replace f(x) by sin(2x) to calculate derivative of sin(2x). Therefore,
f(x) = lim 3[sin(x+h) - sin x] / h
Now, by the trigonometric formula, sin A cos B + cos A sin B = sin (A + B)
f(x) = lim 3[sin x cos h + cos x sin h - sin x] / h
f(x) = lim 3[-sin x(1 - cos h) + cos x sin h] / h
Now, by using the half-angle formula, 1- cos h = 2 sin2 (h / 2), the above equation is written as:
f(x) = 3[(- sin x) { lim [(2 sin2 (h / 2))] / h} + (cos x) {lim (sin h)] / h}
f(x) = 3[(-sin x) [lim (sin(h / 2)) / (h / 2). Lim sin (h / 2)] + (cos x) {lim (sin h)] / h}
As we know,
Lim (sin x / x) = 1, we get
f(x) = 3[- sin x (1. sin (0 / 2)) + cos x (1)]
f(x) = 3[- sin x(0) + cos x]
f(x) = 3cos x
Also use our first principle of derivative calculator to find the derivative of a function by using first definition of derivatives.
Derivative of 3sin(x) by chain rule
To calculate the differentiation of 3sinx, we can use the chain rule since the cosine function can be expressed as a combination of two functions. The chain rule of derivatives states that the derivative of a composite function is equal to the derivative of the outer function multiplied by the derivative of the inner function. The chain rule of derivatives is defined as;
dy / dx = dy / du x du / dx
Where, u=g(x), y=f(u), dy/du is the derivative of f(u) with respect to u and du/dx is the derivative of g(x) with respect to x.
Proof of derivative of 3sin x by chain rule
To prove the derivative of 3sin x by using chain rule, assume that sin x can be written as sin x = cos (π / 2 - x). Using this let us find the differentiation of 3sinx.
We can differentiate 3 sin x using chain rule,
y = - 3sin(π / 2 - x) . d / dx (π / 2 - x)
= - 3sin(π / 2 - x) . (-1)
= 3sin(π / 2 - x)
= 3cos x
Thus, we have derived the formula of derivative of 3sin x by chain rule.
You can also use chain rule calculator with steps and get step by step results.
Derivative of 3sin x using quotient rule
Another method for finding the derivative of 3sin x is using the quotient rule, which is a formula for finding the derivative of a quotient of two functions. Since the sin function is the reciprocal of cosecant, the derivative 3sin x can also be calculated using the quotient rule. The quotient rule is defined as:
d / dx (f / g) = f(x). g(x) -g(x).f(x) / {g(x)}2
Proof of derivative of 3sin x by quotient rule
To prove the derivative of 3sin x, we can write it,
f(x) = 3sin x = 3 / cocec x = u / v
Supposing that u = 31 and v = 3cosec x. Now by quotient rule,
f(x) = (vu - uv) / v2
f(x) = [ccosec x d / dx(3) - 3. d / dx(cosec x)] / (cosec x)2
= [tan x (0) - 1 (-3cosec x cot x)] / cosec2x
= (3cosec x.cot x) / cosec2x
= 3cos x
Hence, we have derived the derivative of 3sin x using the quotient rule of differentiation.
With quotient rule calculator, you can do you calculations instantly.
How to find the differentiation of 3sinx with a calculator?
The easiest way to calculate the derivative of 3sin x is by using an online tool. You can use our derivative calculator with steps free for this. Here, we provide you a step-by-step way to calculate derivatives by using this tool.
Write the function as 3sin x in the enter function box. In this step, you need to provide input value as a function as you have to calculate the derivative of 3sin x.
Now, select the variable by which you want to differentiate 3sin x. Here you have to choose x.
Select how many times you want to differentiate 3sin x. In this step, you can choose 2 for second derivative, 3 for third derivative and so on.
Click on the calculate button. After this step, you will get the derivative of 3sinx within a few seconds.
After completing these steps, you will receive the differentiation of 3sin x within seconds. Using online tools can make it much easier and faster to calculate derivatives, especially for complex functions.
Frequently Asked Questions
What is the first derivative of sin?
The derivative from first principle is the measure of rate of change in a function. The derivative of sin x is denoted by d / dx (sin x) = cos x.
How do you prove the derivative of sinx?
The derivative of sinx can be prove by using first principle of the differentiations. The first principle of differentiation measure the instantaneous rate of change such that;
f(x) = lim f(x+h) - f(x) / h
What are first principles in maths?
Axioms or postulates are terms used in mathematics to describe first principles. Theoretical work is said to be from first principles, or ab initio, in physics and other sciences if it begins directly at the level of established science and does not make assumptions such as empirical model and parameter fitting.