Differentiation Rules

Derivative Rules for Sums, Differences, Constants, and Powers

Introduction

In calculus, understanding the rules of differentiation is crucial for solving various problems. These rules allow us to find the derivative of functions involving sums, differences, constants, and powers.

Individual Derivative Rules

* Power Rule:

If f(x) = xⁿ, then f'(x) = nx^n-1

* Constant Multiple Rule:

If f(x) = c*g(x), where c is a constant, then f'(x) = c*g'(x)

* Sine Function Rule:

If f(x) = sin(x), then f'(x) = cos(x)

Sum and Difference Rules

These rules are applied when combining derivatives of different functions: * Sum Rule:

If f(x) = g(x) + h(x), then f'(x) = g'(x) + h'(x)

* Difference Rule:

If f(x) = g(x) - h(x), then f'(x) = g'(x) - h'(x)

Examples

Using these rules, let's calculate the derivative of fgx fgxgx: * f(x) = fgx fgxgx * f'(x) = (fgx)'(fgxgx) + (fgxgx)'(fgx) * f'(x) = (g(x) + fg(x)) * (fgx) + (fgx) * (g'(x) + fg'(x)) * f'(x) = Fg(x) 1g(x) 2gx sinx + Fg(x) 1g(x) 2gx cosx

Conclusion

The derivative rules for sums, differences, constants, and powers provide the foundation for differentiation in calculus. By understanding these rules, we can simplify the process of finding derivatives and solve a wide range of calculus problems. These rules are essential for students, researchers, and practitioners in various fields that involve calculus.