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Differentiation and integration are two fundamental operations in calculus.

• Differentiation finds the rate of change or slope of a function.

• Integration finds the area under a curve or the accumulated value.

Both play a key role in physics, engineering, economics, and real-life applications. Let’s explore all important formulas with easy explanations.

Why it matters
Calculus uses derivatives in many contexts: motion (velocity), optimization (finding maxima/minima), curve-shape analysis, and more. Having the standard formulas at your fingertips makes solving problems much easier.

1. Basic Rules of Differentiation

These are the building blocks for all other derivative work:

• Constant rule: If $y = c$y=c (where $c$c is a constant), then $-frac{dy}{dx} = 0.$dxdy​=0.

• Power rule: If $y = x^n$y=xn, then $-frac{dy}{dx} = n -cdot x^{-,n-1}.$dxdy​=n⋅xn−1.

• Constant multiple rule: If $y = c -cdot f(x)$y=c⋅f(x), then $y' = c -cdot f'(x).$y′=c⋅f′(x).

• Sum/difference rule: If $y = f(x) -pm g(x)$y=f(x)±g(x), then $y' = f'(x) -pm g'(x).$y′=f′(x)±g′(x).

• Product rule: If $y = f(x)-cdot g(x)$y=f(x)⋅g(x), then

  $$-frac{dy}{dx} = f'(x)-cdot g(x) + f(x)-cdot g'(x).$$dxdy​=f′(x)⋅g(x)+f(x)⋅g′(x).

• Quotient rule: If $y = -frac{f(x)}{g(x)}$y=g(x)f(x)​, then

  $$-frac{dy}{dx} = -frac{-,g(x)-,f'(x) - f(x)-,g'(x)-,}{[g(x)]^2}.$$dxdy​=[g(x)]2g(x)f′(x)−f(x)g′(x)​.

These rules let you handle combinations of functions.

2. Derivatives of Common Functions

Once you know the rules, you also need the derivatives of commonly encountered functions:

Algebraic / Power / Log-exponential functions

• $-frac{d}{dx}(a^x) = a^x -ln a$dxd​(ax)=axlna

• $-frac{d}{dx}(e^x) = e^x$dxd​(ex)=ex

• $-frac{d}{dx}(-ln x) = -frac1x$dxd​(lnx)=x1​

• $-frac{d}{dx}(-log_a x) = -frac1{(-ln a)-,x}$dxd​(loga​x)=(lna)x1​

  
  

Trigonometric functions

• $-frac{d}{dx}(-sin x) = -cos x$dxd​(sinx)=cosx

• $-frac{d}{dx}(-cos x) = --sin x$dxd​(cosx)=−sinx

• $-frac{d}{dx}(-tan x) = -sec^2 x$dxd​(tanx)=sec2x

• $-frac{d}{dx}(-cot x) = --csc^2 x$dxd​(cotx)=−csc2x

• $-frac{d}{dx}(-sec x) = -sec x-,-tan x$dxd​(secx)=secxtanx

• $-frac{d}{dx}(-csc x) = --csc x-,-cot x$dxd​(cscx)=−cscxcotx

  
  

Inverse trigonometric functions

• $-frac{d}{dx}(-sin^{-1} x) = -frac1{-sqrt{1 - x^2}}$dxd​(sin−1x)=1−x2​1​

• $-frac{d}{dx}(-cos^{-1} x) = --frac1{-sqrt{1 - x^2}}$dxd​(cos−1x)=−1−x2​1​

• $-frac{d}{dx}(-tan^{-1} x) = -frac1{1 + x^2}$dxd​(tan−1x)=1+x21​

• Other similar forms for $-sec^{-1} x$sec−1x, $-csc^{-1} x$csc−1x, $-cot^{-1} x$cot−1x.

  
  

Hyperbolic functions (less common in early courses)

• $-frac{d}{dx}(-sinh x) = -cosh x$dxd​(sinhx)=coshx

• $-frac{d}{dx}(-cosh x) = -sinh x$dxd​(coshx)=sinhx

• And similar derivative rules for $-tanh x$tanhx, $-mathrm{csch}-,x$cschx, $-mathrm{sech}-,x$sechx, $-coth x$cothx.

  
  

3. Composite Functions: The Chain Rule

When you have a function inside another function—say $y = f(g(x))$y=f(g(x))—you use the chain rule:

In more extended form: if you have several layers $y = f(h(g(x)))$y=f(h(g(x))), then differentiate each layer from outer to inner:

This rule is essential for many real-world problems and advanced functions.

4. Why Use the Formulas and How to Get Good at Them

• Faster solving: With the standard formulas memorised, you can compute derivatives quickly instead of deriving each one from the limit definition.

• Confidence: You’ll feel more comfortable with algebraic manipulations, especially rules like product/quotient, chain rule, implicit differentiation.

• Deeper understanding: Knowing which rule to apply and why helps you understand what the derivative means (rise over run, slope of tangent, instantaneous rate) rather than blindly applying memorised steps.

• Better in exams: Many exams ask for derivative of combined functions (e.g., $-ln(-sin x)$ln(sinx), $e^{x^2} -cdot -cos x$ex2⋅cosx). If you recognise the inner/outer structure and apply chain/product rules you’ll be more accurate.

Tip: Practice is key. Take a function, identify which rules apply (power + chain, or trig + product, etc), then compute step by step. Also verify your result by thinking: does the sign make sense? Is the domain valid (especially for inverse trig)?

Differentiation Formulas

Integration Formulas

Final Thoughts

Keep a list of these formulas handy. Practice applying them. Start with simple cases (power, trig) and then move to harder ones (chain rule, quotient). With steady practice, you’ll get comfortable extracting derivatives quickly.

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