Question

Find the derivative of y with respect to the given independent variable. Use only LOGARITHMIC differentiation.

47. y=t(t+1)(t+2)

55.Find the derivative of y with respect to theta
y=ln ((cos^2)(theta))

Answer 1

47). y = t(t + 1)(t + 2).

Distribute terms using distributive property:  a( b + c) = ab + ac

y = (t² + t)(t + 2)

FOIL method: the product of two binomials is the sum of the products of the First terms, the Outer terms, the Inner terms and the Last terms.

y = t³ + 2t² + t² + 2t

y = t³ + 3t² + 2t

Apply 'derivative of y with respect to t' each side.

dy/dt = (d/dt)(t³ + 3t² + 2t)

dy/dt = (d/dt)(t³) + (d/dt)(3t²) + (d/dt)(2t)

Derivative of Power Rule: (d/dx)(xⁿ) = nx^n-1

dy/dt =3t² + 3(2t) + 2(1)

dy/dt = 3t² + 6t + 2.

Therefore the derivative of y is 3t² + 6t + 2.

Answer 2

55). y = log[cos²θ]

y = log[cosθ]²

As per the log property, we can move the exponent within the log to the outside.

y = 2 ln(cosθ)

Apply 'derivative of y with respect to theta' each side.

Now we can differentiate, noting that we can leave the constant 2 alone. We still use the chain rule, but we only use it two times instead of three.

Recall: Derivative of Trigonometric Functions (d/dx)(cosx) = - sin(x)

Derivative of Logarithm Functions: (d/dx)[log(x)]=1/x

dy/dx = 2 [ 1/cosθ][-sinθ]

dy/dx = -2[sinθ/cosθ]

Quotient Identities: sinθ/cosθ = tanθ.

dy/dx = - 2tanθ.

Therefore the derivative of y is - 2tanθ.