Property 12
Chain Rule
\(y=f(u)\)
\(u=h(t)\)
\(t=g(x)\)
\(\boxed{\samefrac{dy}{dx}=\samefrac{dy}{du}\cdot\samefrac{du}{dt}\cdot\samefrac{dt}{dx}}\)
1.
\(y=k+3\)
\(k=3t+5\)
\(\Rightarrow \samefrac{dy}{dt}=?\)
2.
\(y=5t^2-8\)
\(t=2m-1\)
\(\Rightarrow \left.\samefrac{dy}{dm}\right|_{m=2}=?\)
3.
\(y=2x-5\)
\(x=t^2-3\)
\(\Rightarrow \left.\samefrac{dy}{dt}\right|_{t=3}=?\)
4.
\(y=2x^2-4\)
\(x=t^3-1\)
\(\Rightarrow \left.\samefrac{dy}{dt}\right|_{t=2}=?\)
5.
\(y=x^2\)
\(x=4t+1\)
\(t=3m-2\)
\(\Rightarrow \left.\samefrac{dy}{dm}\right|_{m=1}=?\)
6.
\(x=4y^2\)
\(y=3t\)
\(t=k^2-1\)
\(\Rightarrow \left.\samefrac{dx}{dk}\right|_{k=2}=?\)
7.
\(x=4t\)
\(t=y^2+2y\)
\(y=3m-1\)
\(\Rightarrow \left.\samefrac{dx}{dm}\right|_{m=1}=?\)