For a certain moving point

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

For a certain moving point, the coordinate

x

depends on time

t

according to

x=\sqrt{4t+1}

. A second quantity is defined by

y=(3x^{2}+2x)^{\tfrac{5}{2}}

. Determine the rate of change

\dfrac{dy}{dt}

when

t=3\,\mathrm{s}

. (i) Obtain

\dfrac{dy}{dx}

in its simplest form. (ii) Find

\dfrac{dx}{dt}

in terms of

t

. (iii) Hence compute the numerical value of

\dfrac{dy}{dt}

t=3\,\mathrm{s}

, writing the final answer with appropriate units.

Related theory

Advanced Differentiation
Differentiation Rules
Derivatives of standard functions via chain rule
Derivatives of composite functions
Continuity and Differentiability

Browse the problem library