Derivatives
GradesPre-Calculus & CalculusDerivativesDifferentiation Rules

Differentiation Rules

📉 Derivatives
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Differentiation Rules

Apply power, product, quotient, and chain rules to differentiate efficiently.

Computing derivatives from the definition is tedious for complex functions. Differentiation rules provide shortcuts derived from the limit definition. These rules cover all elementary functions and their combinations.

ddx[xn]=nxn−1ddx[ex]=exddx[ln⁡x]=1x\frac{d}{dx}[x^n] = nx^{n-1} \qquad \frac{d}{dx}[e^x] = e^x \qquad \frac{d}{dx}[\ln x] = \frac{1}{x}
ddx[sin⁡x]=cos⁡xddx[cos⁡x]=−sin⁡xddx[tan⁡x]=sec⁡2x\frac{d}{dx}[\sin x] = \cos x \qquad \frac{d}{dx}[\cos x] = -\sin x \qquad \frac{d}{dx}[\tan x] = \sec^2 x
Product Rule: ddx[fg]=f′g+fg′\text{Product Rule: } \frac{d}{dx}[fg] = f'g + fg'
Quotient Rule: ddx[fg]=f′g−fg′g2\text{Quotient Rule: } \frac{d}{dx}\left[\frac{f}{g}\right] = \frac{f'g - fg'}{g^2}
Chain Rule: ddx[f(g(x))]=f′(g(x))⋅g′(x)\text{Chain Rule: } \frac{d}{dx}[f(g(x))] = f'(g(x))\cdot g'(x)
🔧Chain Rule in Action
Differentiate y = sin(3x²). Let u = 3x², so y = sin(u). dy/dx = cos(u) · 6x = 6x cos(3x²).
ddx[sin⁡(3x2)]=cos⁡(3x2)⋅6x=6xcos⁡(3x2)\frac{d}{dx}[\sin(3x^2)] = \cos(3x^2) \cdot 6x = 6x\cos(3x^2)
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Remember This!

For the chain rule: identify the outer function and inner function. Differentiate the outer function (leaving the inner alone), then multiply by the derivative of the inner function.

✏️ Try It!

What is the derivative of f(x) = x⁴ − 5x² + 2x − 7?

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