Differentiation rules are the building blocks of calculus, allowing us to find for various functions. These rules, including the , , , and , give us tools to tackle complex mathematical problems.
By combining these rules, we can differentiate even the most intricate functions. Understanding how to apply these rules in different scenarios is crucial for solving real-world problems involving rates of change, , and more.
Differentiation Rules
Power rule for polynomial differentiation
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Power rule: dxdxn=nxn−1 multiplies coefficient by exponent and subtracts 1 from exponent (x3→3x2)
: dxdc=0 states derivative of constant is always 0 (5→0)
: dxd(af(x)+bg(x))=adxdf(x)+bdxdg(x) differentiates each term separately, multiplies each derivative by its coefficient, and adds results (3x2+2x→6x+2)
Finding derivative of outer function, f, with respect to its input
Finding derivative of inner function, g, with respect to x
Multiplying the two derivatives together
Generalized chain rule: dxdf(g(h(x)))=f′(g(h(x)))g′(h(x))h′(x) applies chain rule successively for each nested function and multiplies derivatives of all functions in composition
Example: dxdsin(x2)=cos(x2)⋅2x
Application of Differentiation Rules
Combine differentiation rules to find derivatives of complex functions
Identify structure of complex function as sum, product, quotient, or composition of functions
Apply appropriate differentiation rules based on function's structure:
Linearity of differentiation for sums of functions
Product rule for products of functions
Quotient rule for ratios of functions
Chain rule for
Simplify resulting derivative expression if necessary