Trigonometric Calculus Tool

Trigonometric Derivative Calculator

Differentiate trigonometric expressions containing sums, differences, multiplication, division, powers, parentheses, coefficients and composite functions.

Try an example:

sin(x) * cos(x) sin(x) / x 2sin(3x) + 4cos(x) x^2 * sin(x) (sin(x) + cos(x)) / x sin(x)^2 sin(2x) * cos(3x) + tan(x) 3sec(2x) / x

Derivative

\[ 2 \cdot \sin\left(x\right) \cdot \cos\left(x\right) \]

In readable form:

2 * sin(x) * cos(x)

Expression

\[ \sin\left(x\right)^{2} \]

Differentiation Steps

Step 1: Power Rule

For u^n, the power rule is d/dx[u^n] = n·u^(n−1)·u'.

Here u = sin(x) and n = 2.

Final Result

\[ \boxed{ \frac{d}{dx} \left( \sin\left(x\right)^{2} \right) = 2 \cdot \sin\left(x\right) \cdot \cos\left(x\right) } \]

What Is a Trigonometric Derivative?

A trigonometric derivative is the derivative of a function involving trigonometric functions such as sine, cosine, tangent, cotangent, secant and cosecant.

For simple functions such as \(f(x)=\sin(x)\), the derivative follows a standard formula. For more complicated expressions, several differentiation rules may need to be combined.

\[ \frac{d}{dx}\sin(x)=\cos(x) \] \[ \frac{d}{dx}\cos(x)=-\sin(x) \] \[ \frac{d}{dx}\tan(x)=\sec^2(x) \]

This calculator can combine these rules automatically when an expression contains multiple trigonometric terms.

Supported Trigonometric Derivatives

Sine

\[ \frac{d}{dx}\sin(x)=\cos(x) \]

Cosine

\[ \frac{d}{dx}\cos(x)=-\sin(x) \]

Tangent

\[ \frac{d}{dx}\tan(x)=\sec^2(x) \]

Cotangent

\[ \frac{d}{dx}\cot(x)=-\csc^2(x) \]

Secant

\[ \frac{d}{dx}\sec(x) = \sec(x)\tan(x) \]

Cosecant

\[ \frac{d}{dx}\csc(x) = -\csc(x)\cot(x) \]

Product Rule

When two differentiable expressions are multiplied, the product rule is required.

\[ \frac{d}{dx}[u(x)v(x)] = u'(x)v(x)+u(x)v'(x) \]

For example, consider:

\[ f(x)=\sin(x)\cos(x) \]

Applying the product rule gives:

\[ f'(x) = \cos(x)\cos(x) - \sin(x)\sin(x) \]

The calculator can apply the product rule automatically to expressions such as \(x\sin(x)\), \(\sin(x)\cos(x)\), \(2\sin(x)\tan(x)\), and more.

Quotient Rule

When one expression is divided by another, the quotient rule is used.

\[ \frac{d}{dx} \left[ \frac{u}{v} \right] = \frac{u'v-uv'}{v^2} \]

For example:

\[ f(x)=\frac{\sin(x)}{x} \]

Therefore:

\[ f'(x) = \frac{x\cos(x)-\sin(x)}{x^2} \]

Division involving trigonometric expressions is also supported, such as \(\sin(x)/\cos(x)\), \(\sin(2x)/(x^2+1)\), and similar expressions.

Chain Rule

The chain rule is required when a trigonometric function contains another function of \(x\).

\[ \frac{d}{dx}f(g(x)) = f'(g(x))g'(x) \]

For example:

\[ f(x)=\sin(2x) \] \[ f'(x)=2\cos(2x) \]

The same principle works with more complicated inner expressions.

\[ \frac{d}{dx}\cos(x^2) = -2x\sin(x^2) \]

Power Rule

The calculator supports constant powers of expressions, including powers of trigonometric functions.

\[ \frac{d}{dx}[u(x)^n] = n\,u(x)^{n-1}u'(x) \]

For example:

\[ f(x)=\sin^2(x) \] \[ f'(x) = 2\sin(x)\cos(x) \]

Expressions such as \(x^2\sin(x)\), \(\sin^3(x)\), and \((\cos(x)+1)^2\) can therefore be differentiated.

Multiplication and Division Examples

Unlike a calculator that only accepts one trigonometric term, this tool parses the complete expression before differentiating.

Input	Rule Used
`sin(x) * cos(x)`	Product Rule
`x * sin(x)`	Product Rule
`sin(x) / x`	Quotient Rule
`sin(x) / cos(x)`	Quotient Rule
`2sin(x)`	Constant Multiple Rule
`3cos(2x)`	Constant Multiple + Chain Rule
`sin(x)^2`	Power + Chain Rule
`sin(x)*cos(2x)+tan(x)`	Product + Chain + Sum Rules

How to Enter Expressions

You can enter multiplication explicitly with *, or use common implicit multiplication.

2sin(x) means \(2\sin(x)\)
3x means \(3x\)
xsin(x) means \(x\sin(x)\)
sin(x)cos(x) means \(\sin(x)\cos(x)\)
(x+1)sin(x) means \((x+1)\sin(x)\)

You can also use explicit multiplication:

\[ 2*\sin(x) \] \[ x*\cos(x) \] \[ \frac{\sin(x)}{\cos(x)} \]

Supported Expression Structure

The parser recognizes mathematical expressions using the following structure:

\[ \text{expression} = \text{sum/difference of products and quotients} \]

Parentheses can be nested, allowing expressions such as:

\[ \frac{\sin(x)+\cos(x)} {x^2+1} \] \[ x^2\sin(2x) \] \[ (\sin(x)+1)^2 \]

Constant powers are supported. Variable exponents such as \(x^x\) are intentionally not expanded by this calculator because they require logarithmic differentiation.

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Frequently Asked Questions

Can this calculator differentiate sin(x) + cos(x)?

Yes. It parses each term and applies the sum rule:

\[ \frac{d}{dx} [\sin(x)+\cos(x)] = \cos(x)-\sin(x) \]

Can it differentiate products?

Yes. Multiplication is handled using the product rule. For example, sin(x)*cos(x) is supported.

Can it differentiate fractions?

Yes. Division is handled using the quotient rule. For example, sin(x)/x and sin(x)/cos(x) are supported.

Can I use coefficients such as 2sin(x)?

Yes. The parser recognizes implicit multiplication, so 2sin(x) is interpreted as \(2\sin(x)\).

Can I use sin(2x)?

Yes. The derivative is calculated using the chain rule:

\[ \frac{d}{dx}\sin(2x) = 2\cos(2x) \]

Can I use powers such as sin(x)^2?

Yes. Constant powers are supported using the power rule and chain rule.

Can I use parentheses?

Yes. Nested parentheses are supported, which allows more complicated products, quotients and composite functions.

Does it support x^x?

No. The calculator supports constant exponents such as \(x^2\), \(x^3\), and \(\sin^2(x)\). A variable exponent such as \(x^x\) requires logarithmic differentiation and is outside the current symbolic parser.