Derivative of Sin x

The derivative of sin x formula is one of the formulas of differentiation. There are specific formulas in differentiation to find the derivatives of different types of functions. All these formulas are basically derived from the limit definition of the derivative (which is called derivative by the first principle). Here also we are going to prove the derivative of sin x to be -cos x using the first principle.

Let us learn how to do the differentiation of sin x along with a few examples. Also, let us study the graph of sin x and the derivative of sin x.

The derivative of sin x with respect to x is cos x. It is represented as d/dx(sin x) = cos x (or) (sin x)' = cos x. i.e., the derivative of sine function of a variable with respect to the same variable is the cosine function of the same variable. i.e.,

• d/dy (sin y) = cos y
• d/dθ (sin θ) = cos θ

Derivative of Sin x Formula

The derivative of sin x is cos x. We are going to prove this in each of the following methods.

• By the first principle
• By chain rule
• By quotient rule

The limit definition of the derivative (first principle) is used to find the derivative of any function. We are going to use the first principle to find the derivative of sin x as well. For this, let us assume that f(x) = sin x to be the function to be differentiated. Then f(x + h) = sin(x + h). Now, by the first principle, the limit definition of the derivative of a function f(x) is,

f'(x) = limₕ→₀ [f(x + h) - f(x)] / h

Substituting f(x) = sin x and f(x + h) = sin(x + h) here,

f'(x) = limₕ→₀ [sin(x + h) - sin x] / h

We can evaluate this limit in two methods.

Method 1

By one of the trigonometric formulas, sin C - sin D = 2 cos [(C + D)/2] sin [(C - D)/2]. Applying this,

f'(x) = limₕ→₀ [2 cos[(x + h + x)/2] sin[(x + h - x)/2] ] / h

= limₕ→₀ [2 cos[(2x + h)/2] sin (h/2) ] / h

= limₕ→₀ [cos[(2x + h)/2] · limₕ→₀ [sin (h/2) ] / (h/2)]

As h → 0, (h/2) → 0. So

f'(x) = limₕ→₀ [cos[(2x + h + x)/2] · lim₍ₕ/₂₎→₀ [sin (h/2) ] / (h/2)]

Using limit formulas, lim ₓ→₀ (sin x/x) = 1. So

f'(x) = [cos[(2x + 0)/2] · (1) = cos (2x/2) = cos x

Thus, we have proved that the derivative of sin x is cos x.

Method 2

By sum and difference formulas,

sin (A + B) = sin A cos B + cos A sin B

Using this,

f'(x) = limₕ→₀ [sin x cos h + cos x sin h - sin x] / h

= limₕ→₀ [ - sin x (1- cos h) + cos x sin h] / h

= limₕ→₀ [ - sin x (1 - cos h)]/h + limₕ→₀ (cos x sin h)/h

= -sin x limₕ→₀ (1 - cos h)/h + (cos x) limₕ→₀ sin h/h

Using half angle formulas, 1 - cos h = 2 sin²(h/2).

f'(x) = -sin x limₕ→₀ (2 sin²(h/2))/h + (cos x) limₕ→₀ sin h/h

= -sin x [limₕ→₀ (sin(h/2))/(h/2) · limₕ→₀ sin (h/2)] + (cos x) (limₕ→₀ sin h/h)

We know that lim ₓ→₀ (sin x/x) = 1.

f'(x) = -sin x (1 · sin(0/2)) + cos x (1)

= -sin x(0) + cos x (From trigonometric table, sin 0 = 0)

= cos x

Hence we have derived that the derivative of sin x is cos x.

By chain rule of differentiation, d/dx(f(g(x)) f'(g(x)) · g'(x). So to find the derivative of sin x using the chain rule, we must write it as a composite function. Using one of the trigonometric formulas, we can write sin x as, sin x = cos (π/2 - x). Using this let us find the derivative of y = sin x (or) cos (π/2 - x) (by co-function identity).

Using chain rule,

y' = - sin(π/2 - x) · d/dx (π/2 - x) (as the derivative of cos x is - sin x)

= - sin(π/2 - x) · (-1)

= sin(π/2 - x)

= cos x (By one of the trigonometric formulas).

Thus, we have derived the formula of derivative of sin x by chain rule.

The quotient rule says d/dx (u/v) = (v u' - u v') / v². So to find the differentiation of sin x using the quotient rule, we have to write sin x as a fraction. We know that sin is the reciprocal of the cosecant function (csc). i.e., y = sin x = 1/(csc x). Then by chain rule,

y' = [csc x · d/dx(1) - 1 · d/dx(csc x)] / csc²x

= [csc x (0) - 1 (-csc x cot x)] / csc²x (as the derivative of csc x is -csc x cot x]

= (cot x) / (csc x)

= [(cos x)/(sin x)] / [1/sin x]

= cos x

Therefore, the derivative of sin is cos x and is proved by using the quotient rule.

The following graph shows the graphs of sin x and its derivative (cos x). We know that a function has a maximum/minimum at a point where its derivative is 0. We can observe in the following graph that wherever sin x is maximum/minimum, cos x is zero at all such points. This way, we can prove that the derivative of sin x is cos x graphically.

sin(u(x)) is a composite function and hence it can be written as sin(u(x)) = f(g(x)) where g(x) = u(x) and f(x) = sin x. Then g'(x) = u'(x) and f'(x) = cos x. We know that the derivative of a composite function is found by using the chain rule. By using chain rule,

d/dx (sin(u(x)) = f'(g(x)) · g'(x)

Since f'(x) = cos x and g(x) = u(x), we have f'(g(x)) = cos (u(x)). So

d/dx (sin(u(x)) = cos (u(x)) · u'(x)

Therefore, the derivative of the composite function sin(u(x)) is cos (u(x)) · u'(x).

Important Notes on Derivative of Sin x:

Here are some important points to note from the differentiation of sin x.

• The derivative of sin x with respect to x is cos x.
• The derivative of sin u with respect to x is, cos u · du/dx.
• Sin x is maximum at x = π/2, 5π/2, .... and minimum at x = 3π/2, 7π/2, ...
  At all these points, the derivative of sin x is 0.
  i.e., at all these points cos x = 0.
• Derivative of cos x is NOT sin x, but it is -sin x.

☛ Related Topics:

Here are the topics that you may be interested in the derivative of sin x.

• Derivative Calculator
• Second Derivative Calculator
• Product Rule
• Derivative of ln x

What is the Differentiation of Sin x?

The differentiation of sin x is cos x.. i.e., the derivative of sin x with respect to x is cos x. It is mathematically written as d/dx(sin x) (or) (sin x)' = cos x.

What is the Difference Between Integral and Derivative of Sin x?

• The derivative of sin x is cos x and is written as d/dx (cos x).
• The integral of sin x is -cos x and is written as  ∫ sin x dx = -cos x + C.

How do You Find the Derivative of Sin x?

The derivative of sin x is found by the first principle. For this we have to substitute f(x) = sin x in the limit definition of the derivative (first principle) which says f'(x) = limₕ→₀ [f(x + h) - f(x)] / h and then simplify. For more clear proof, click here.

How to Derive the Derivative of Sin x Formula Graphically?

If we observe the graphs of sin x and cos x, at all the points at which sin x has either maximum or minimum, cos x is 0. It means that at all critical points of sin x, cos x = 0. Therefore, the derivative of sin x is cos x. For more detailed proof (along with graph), you can visit the "Graph of Sin x and Derivative of Sin x" section of this page.

Why Differentiation of Sin x is Cos x?

The derivative of sin x is cos x and it can be proved using the derivative formula of first principle. For detailed proof, you can visit the "Derivative of Sin x Proof by First Principle" section of this page.

Is the Derivative of Sin x Same as the Derivative of Sin Inverse x?

No, the derivative of sin x is NOT same as that of sin inverse x.

• The derivative of sin x is cos x. i.e., d/dx(sin x) = cos x.
• The derivative of sin inverse is 1/√1 - x². i.e., d/dx(sin^-1x) = 1/√1 - x².

What is the Derivative of Sin √x?

The derivative of sin is cos. Using the chain rule, d/dx(sin √x) = cos √x · d/dx(√x) = cos√x · (1/2√x) = (cos√x)/(2√x) (because the derivative of √x is 1/2√x).

What is the Derivative of Sin 3x?

The derivative of the sine function is the cosine function. Using this and chain rule, d/dx(sin 3x) = cos 3x · d/dx(3x) = cos 3x · (3) = 3 cos 3x. Thus, the derivative of sin 3x is 3 cos 3x.

How to Prove the Derivative of Sin x Formula by First Principle?

To derive the formula for the differentiation of sin x using first principle, assume that f(x) = sin x. Substituting this in f'(x) = limₕ→₀ [f(x + h) - f(x)] / h, we get f'(x) = limₕ→₀ [sin(x + h) - sin x] / h = limₕ→₀ [2 cos[(x + h + x)/2] sin[(x + h - x)/2] ] / h = limₕ→₀ [cos[(2x + h + x)/2] · lim₍ₕ/₂₎→₀ [sin (h/2) ] / (h/2)] = [cos[(2x + 0)/2] · (1) = cos x. Therefore, the derivative of sin x by first principle is cos x.

What is the Derivative of Sin x^2?

We know that the derivative of sin x is cos x. So the derivative of sin x² using this and using the chain rule is cos x²· d/dx(x²). By power rule, d/dx (x²) = 2x. So, the derivative of sin x² is 2x cos x².