Calculus Help!!!

Question

Differentiate the following:

(a) f(t) = e^(5t−π) sqrt(2 − t)

(b) f(x) = ax2 + b/ cx^3 − k

(a, b, c, k are nonzero constants)

Answer 1

Looking at the first function ...

f(t) = e^5t-π * sqrt(2 - t)

Using the product rule... (u(x) * v(x))' = u'(x) * v(x) + u(x) * v'(x)

define u(t) = e^5t-π

define v(t) = sqrt(2 - t)

u'(t) = 5e^5t-π

v'(t) = -1/sqrt(2 - t)

Therefore f'(t) = 5e^5t-π * sqrt(2 - t) - e^5t-π / sqrt(2 - t)

Simplifying --> f'(t) = e^5t-π / sqrt(2 - t) * [ 5*(2 - t) - 1 ]

f'(t) = e^5t-π / sqrt(2 - t) * [ 9 - 5t ]

Looking at the second function...

f(x) = ax² + b / (cx³ - k)

Using the quotient rule (u(x)/v(x))' = [ u'(x) * v(x) - u(x) * v'(x) ] / (v(x))²

define u(x) = ax² + b

define v(x) = cx³ - k

u'(x) = 2ax

v'(x) = 3cx²

Therefore f'(x) = [ 2ax * (cx³ - k) - 3cx² * (ax² + b) ] / (cx³ - k)²

f'(x) = (2acx⁴ - 2akx - 3acx⁴ - 3bcx²) / (cx³ - k)²

f'(x) = -x * (acx³ + 3bcx + 2ak) / (cx³ - k)²