Derivative of Sinx Cosx

The derivative of sinx cosx is equal to cos2x. Differentiation of sinx cosx is the process of finding the derivative of sinx cosx with respect to x which gives the rate of change in the function sinx cosx with respect to the variable x. It is mathematically written as d(sinx cosx)/dx = cos2x = cos²x - sin²x. We can find the derivative of sinx cosx using different methods of differentiation such as the first principle of derivatives and product rule of differentiation.

In this article, we will calculate the derivative of sinx cosx using various differentiation methods. We will solve various examples using the differentiation of sinx cosx for a better understanding of the concept.

The derivative of sinx cosx is equal to the cos2x. The process of finding the derivatives in calculus is called differentiation. It gives the instantaneous rate of change in a function with respect to a variable. We can calculate the derivative of sinx cosx using the first principle of differentiation, that is, the definition of limits and the product rule of differentiation. To find this derivative, we use the formula for the derivative of cosx and the derivative of sinx. Let us now go through the formula for the derivative of sinx cosx.

The formula for the derivative of sinx cosx is given by, d(sinx cosx)/dx (OR) (sinx cosx)' = cos2x (OR) cos²x - sin²x. The derivative of a function is the slope of the tangent to the function at the point of contact. The image below shows the formula for the derivative of sinx cosx. It is equal to cos2x when calculated using the first principle and it is equal to cos²x - sin²x when calculated using the product rule.

Now that we know that the derivative of sinx cosx is equal to cos2x, we will prove it using the first principle of differentiation. To find the derivative of sinx cosx, 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. The derivative of a function f(x) is given by the formula f'(x) = lim_h→0[f(x + h) - f(x)] / h. We will use the following trigonometric and limit formulas to solve the derivative:

• sin(A+B) = sinAcosB + cosAsinB
• cos(A+B) = cosAcosB - sinAsinB
• lim_x→0 (sinx)/x = 1
• lim_x→0 (cosx - 1)/x = 0
• cos2A = cos²A - sin²A

Using the above formulas, we have

(sinx cosx)' = lim_h→0[sin(x + h) cos(x + h) - sinx cosx] / h

= lim_h→0[(sinx cosh + sinh cosx) (cosx cosh - sinx sinh) - sinx cosx] / h

= lim_h→0[sinx cos x cos²h + sinh cosh cos²x - sin²x sinh cosh - cosx sinx sin²h - sinx cosx] / h

= lim_h→0[(sinx cos x cos²h - cosx sinx sin²h - sinx cosx) + sinh cosh cos²x - sin²x sinh cosh] / h

= lim_h→0[(sinx cos x cos²h - cosx sinx sin²h - sinx cosx) + sinh cosh cos²x - sin²x sinh cosh] / h

= lim_h→0[sinx cos x (cos²h - sin²h - 1) + sinh cosh (cos²x - sin²x)] / h

= lim_h→0[sinx cos x (cos2h - 1) + sinh cosh cos2x] / h --- [Using Cos2A Formula]

= lim_h→0[sinx cos x (cos2h - 1)] / h + lim_h→0 [sinh cosh cos2x] / h

= 2 sinx cos x lim_h→0(cos2h - 1) / 2h + cos2x lim_h→0 sinh / h × lim_h→0 cosh

= 2 sinx cosx × 0 + cos2x × 1 × 1

= 0 + cos2x

= cos2x

Hence, we have proved that the derivative of cos2x is equal to cos2x.

In this section, we will evaluate the derivative of sinx cosx using the product rule of differentiation. For function h(x) given as the product of two functions f(x) and g(x), that is, h(x) = f(x) g(x), then its derivative is given by h'(x) = f'(x) g(x) + f(x) g'(x). For the function sinx cosx, f(x) = sinx and g(x) = cosx. Then, the derivative of sinx cosx is,

d(sinx cosx)/dx = (sinx)' cosx + sinx (cosx)'

= cosx cosx - sinx (-sinx) --- [Because the derivative of sinx is cosx and the derivative of cosx is -sinx]

= cos²x - sin²x

= cos2x --- [Using cos2x formula]

Hence, the derivative of sinx cosx is equal to cos2x.

Important Notes on Derivative of Sinx Cosx

• The derivative of sinx cosx is equal to cos2x.
• We can evaluate the differentiation of sinx cosx using the power rule and first principle method.
• We can write the derivative of sinx cosx as (sinx cosx)' = cos2x OR cos²x - sin²x

☛ Related Topics:

• Derivative of Sin3x
• Derivative of cos3x
• Derivative of xsinx

What is Derivative of Sinx Cosx in Calculus?

The derivative of sinx cosx is equal to cos2x. It is equal to the rate of change in the function sinx cosx with respect to the variable x.

What is the Formula of Derivative of Sinx Cosx?

The formula for the derivative of sinx cosx is given by, d(sinx cosx)/dx (OR) (sinx cosx)' = cos2x (OR) cos²x - sin²x. We can find the derivative of sinx cosx using different methods of differentiation such as the first principle of derivatives and product rule of differentiation.

How to Prove the Derivative of Sinx Cosx?

We can find the derivative of sinx cosx using the first principle of derivatives, that is, the definition of limits and using the product rule of differentiation.

What is the Second Derivative of Sinx Cosx?

The second derivative of sinx cosx is equal to -2 sin2x which can be calculated by differentiating the first derivative of sinx cosx. It is obtained as d²(sinx cosx)/dx² = d(cos2x)/dx = -2 sin2x.

How is the Derivative of Sinx Cosx Equal to Cos²x - Sin²x?

The derivative of sinx cosx is equal to cos²x - sin²x using the product rule of differentiation. We have d(sinx cosx)/dx = (sinx)' cosx + sinx (cosx)' = cos²x - sin²x.

What is the Derivative of Sinx Cosx Using the Product Rule?

The derivative of sinx cosx is equal to cos²x - sin²x using the product rule of differentiation.