14.3 Standard Derivatives and Chain Rule

2026 📋 Syllabus Objectives

By the end of these notes, you will be able to:

  • State and use the derivatives of the standard functions: xnx^n, sin⁡x\sin x, cos⁡x\cos x, tan⁡x\tan x, exe^x, and ln⁡x\ln x
  • Differentiate expressions involving constant multiples and sums of these functions
  • Differentiate composite functions (a function inside another function) using the chain rule
  • Apply all of the above to exam-style problems

⚠️ Important: For all trigonometric (trig) functions in this topic, angles are always measured in radians, not degrees.

📖 Main Content

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Part 1: What Is a Derivative?

The derivative of a function tells you the rate of change — in other words, how quickly the output of the function changes as the input changes. When you have a curve y=f(x)y = f(x), the derivative dydx\frac{dy}{dx} gives you the gradient (steepness) of the curve at any point.

Differentiation is the process of finding the derivative.

Part 2: Standard Derivatives — The Core Table

These are the building blocks of differentiation. You must memorise these results. Think of them as your "toolkit" — every more complex problem is built from these.

Function yyDerivative dydx\dfrac{dy}{dx}
xnx^n (where nn is any rational number)nxn−1nx^{n-1}
sin⁡x\sin xcos⁡x\cos x
cos⁡x\cos x−sin⁡x-\sin x
tan⁡x\tan xsec⁡2x\sec^2 x
exe^xexe^x
ln⁡x\ln x1x\dfrac{1}{x}

📝 Plain-English Explanations:

  • xnx^n means xx raised to the power nn, where nn can be any fraction or integer (e.g., x2x^2, x−3x^{-3}, x1/2x^{1/2}).
  • exe^x is a special exponential function — its derivative is itself! (The number e≈2.718e \approx 2.718.)
  • ln⁡x\ln x means the natural logarithm of xx (logarithm to base ee).
  • sec⁡2x\sec^2 x means 1cos⁡2x\frac{1}{\cos^2 x}, but you just need to know that this is the derivative of tan⁡x\tan x.
  • All trig functions use radians.

Part 3: Differentiating xnx^n — The Power Rule

The rule for xnx^n is:

ddx(xn)=nxn−1\frac{d}{dx}(x^n) = nx^{n-1}

In plain English: bring the power down to the front, then reduce the power by 1.

This works for any rational value of nn — whole numbers, fractions, and negative numbers.

✏️ Examples:

Example 1: Differentiate y=x5y = x^5

dydx=5x5−1=5x4\frac{dy}{dx} = 5x^{5-1} = 5x^4

Example 2: Differentiate y=x−3y = x^{-3} (a negative power)

dydx=−3x−3−1=−3x−4\frac{dy}{dx} = -3x^{-3-1} = -3x^{-4}

Example 3: Differentiate y=x1/2=xy = x^{1/2} = \sqrt{x} (a fractional power)

dydx=12x1/2−1=12x−1/2=12x\frac{dy}{dx} = \frac{1}{2}x^{1/2 - 1} = \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}}

💡 Tip: Always rewrite roots and fractions as powers before differentiating.

  • x=x1/2\sqrt{x} = x^{1/2}
  • 1x3=x−3\frac{1}{x^3} = x^{-3}
  • 1x=x−1/2\frac{1}{\sqrt{x}} = x^{-1/2}

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