Calculus Formulas

About Calculus Formulas

Here are some basic calculus formulas for some common functions' derivatives and integrals. The derivative with respect to x is denoted by d/dx, and the integral of a function, f, with respect to x is denoted by f(x)dx. Furthermore, because all of the integrals are indefinite, they all have a +C constant term. Because a constant's derivative is zero, the anti-derivative of a function might include constant terms.

Derivative Rules

1. Exponential/Logarithmic Functions:
   1. d(e^x)/dx=e^x
   2. d(a^x)/dx=a^x⋅ln(a)
   3. d(ln(x))/dx=1/x
   4. d(log_ax)/dx=1/xln(a)
2. Trigonometric Functions:
   1. d(sin(x))/dx=cos(x)
   2. d(cos(x))/dx=−sin(x)
   3. d(tan(x))/dx=sec²(x)
   4. d(cosec(x))/dx=−cosec(x)cot(x)
   5. d(sec(x))/dx=sec(x)tan(x)
   6. d(cot(x))/dx=−cosec²(x)
3. Inverse Trigonometric Functions:
   1. d(sin⁻¹(x))/dx=1/(1−x²)
   2. d(cos⁻¹(x))/dx=−1/(1−x²)
   3. d(tan^-1x)/dx=1/(1+x²)
   4. d(csc^-1x)/dx=−1/x √(x²−1)
   5. d(sec^-1x)/dx=1/x√(x²−1)
   6. d(cot^-1x)/dx=−1/(1+x²)
4. These are just a few of the derivative rules for common functions that arise in differential calculus.
   1. Integral Rules
   2. ∫(1/x)dx=ln|x|+C
   3. ∫e^xdx=e^x+C
   4. ∫a^xdx=a^xln(a)+C
   5. ∫sin(x)dx=−cos(x)+C
   6. ∫cos(x)dx=sin(x)+C
   7. ∫tan(x)dx=ln|sec(x)|+C
   8. ∫cot(x)dx=ln|sin(x)|+C
   9. ∫sec(x)dx=ln|sec(x)+tan(x)|+C
   10. ∫csc(x)dx=ln|csc(x)−cot(x)|+C

These are just a few of the many different integral rules for disparate functions that may arise in integral calculus.

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