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EXAMPLES OF DERIVATIVES

Determine the derivative of f(x) = x² using the definition of the derivative.

(1) f’(x) = lim_h → 0(f(x + h) − f(x))/(h) = lim_h → 0((x + h)² − x²)/(h) = lim_h → 0(x² + 2xh + h² − x²)/(h) = lim_h → 02x + h = 2x

Determine the derivative of f(x) = x² using the power rule.

(2) f’(xⁿ) = nx^n − 1 → f’(x²) = 2x¹ = 2x

It is clear that calculating the derivative by the power rule gives the same result as calculating it using the definition of the derivative.

Plot the derivative of sinx + 1 for 0 ≤ x ≤ 2π
Using the sum rule and the derivative of sinx:
(3) (af + bg)’ = af’ + bg’ → (sinx + 1)’ = (sinx)’ + (1)’ = cosx

Figure 1 Plot of the derivative of sinx + 1

Determine the derivative f(x) where f = e^cosx.
Let k(x) = cosx, f(k) = e^k. Using the chain rule:

(4) f’(x) = f’(k)⋅k’(x) = e^cosx⋅( − sinx) = − e^cosxsinx

Determine the derivative of xtanx.
Use the product rule:

(5) (fg)’ = gf’ + fg’ → (xtanx)’ = tanx + xsec²x