Answer to Question #253812 in Calculus for yobro

Question

In each part, compute f

′

, f

′′

, f

′′′ and then state the formula for nth derivative f

(n)

(a) f(x) = e

2x

(b) f(x) = sin3x

(c) f(x) = lnx

Draw the graph of first three derivatives of f(x).

Answer 1

$(a)\ f(x) = e^{2x}\\ f'(x) = 2e^{2x}\\ f''(x) = 4e^{2x}\\ f'''(x) = 8e^{2x}\\ f^{(n)}(x) = 2^ne^{2x}$

graph of the first derivative

graph of the second derivative

graph of the third derivative

$(b)\ f(x) = sin3x\\ f'(x) = 3cos3x\\ f''(x) = -9sin3x\\ f'''(x) = -27cos3x\\ f^{(n)}(x) =3^n\sin\bigl(x+\tfrac{n\pi} 2\bigr).$

graph of the first derivative

graph of the second derivative

graph of the third derivative

$(c)\ f(x) = ln(x)\\ f'(x) = 1/x\\ f''(x) = -1/x²\\ f'''(x) = 2/x³\\ f^{(n)}(x) = \dfrac {({n - 1})!( {-1})^{n - 1} } {x^n}$

graph of the first derivative

graph of the second derivative

graph of the third derivative$.$