Derivative of sin(x²)
2026-02-28 23:51 Diff
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We can derive the derivative of sin(x²) using proofs. To show this, we will use trigonometric identities along with the rules of differentiation.

There are several methods we use to prove this, such as:

By First Principle

Using Chain Rule

Using Product Rule

We will now demonstrate that the differentiation of sin(x²) results in 2x cos(x²) using the above-mentioned methods:

Using Chain Rule

To prove the differentiation of sin(x²) using the chain rule, We use the formula: y = sin(u), where u = x² Then dy/du = cos(u) and du/dx = 2x

Using the chain rule: dy/dx = dy/du * du/dx dy/dx = cos(x²) * 2x

Thus, d/dx (sin(x²)) = 2x cos(x²).

Using First Principle The derivative of sin(x²) can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

To find the derivative of sin(x²) using the first principle, we will consider f(x) = sin(x²).

Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1)

Given that f(x) = sin(x²), we write f(x + h) = sin((x + h)²).

Substituting these into equation (1), f'(x) = limₕ→₀ [sin((x + h)²) - sin(x²)] / h

Using trigonometric identities and limit properties, and simplifying, f'(x) = 2x cos(x²).

Using Product Rule

The product rule is not directly applicable to sin(x²) as it is not a product of functions. But for functions involving multiplication of derivatives, the product rule can be useful.