Let 𝑓(𝑥)=5(sin(𝑥))^𝑥. Find 𝑓′(3).

Question

Let 𝑓(𝑥)=5(sin(𝑥))^𝑥. Find 𝑓′(3).

Answer 1

Use logarithmic differentiation:

y = 5(sin(x))^x

ln(y) = ln(5(sin(x))^x)

ln(y) = ln(5) + xln(sin(x))

differentiate both sides:

1/y•(dy/dx) = 0 + (1)(ln(sin(x)) + x•(1/sin(x))•cos(x)

1/y•(dy/dx) = ln(sin(x)) + xcot(x)

dy/dx = y[ln(sin(x)) + xcot(x)]

dy/dx = (5(sin(x))^x)[ln(sin(x)) + xcot(x)]

plugging in x = 3 we get f '(3) = (5(sin(3))³)(ln(sin(3)) + 3cot(3))

f '(3) = 5•0.00281(ln(0.14112) - 21.04576)

f '(3) = 0.01405(-23.0039)

f '(3) = -0.323249