IS curve, interest rate and consumption
The trick is to apply the total derivative.
Going from $Y = C(Y-T(Y)) + I(r) + G + NX(Y)$, we have:
$$
\operatorname{dY} = \underbrace{\frac{\partial C(\cdot)}{\partial Y}}_{\equiv C_Y &gt; 0} \cdot \operatorname{dY} + \underbrace{\frac{\partial I(\cdot)}{\partial r}}_{\equiv I_r &lt; 0} \cdot \operatorname{dr} \therefore \operatorname{dY} - C_Y \cdot \operatorname{dY} = I_r \cdot \operatorname{dr} \Leftrightarrow \boxed{\boxed{\frac{\operatorname{dr}}{\operatorname{dY}}\bigg\vert_{IS} = \frac{1-C_Y}{I_r} &lt; 0}}
$$
Now, with the new equation, we have:
$$
\operatorname{dY} = \underbrace{\frac{\partial C(\cdot)}{\partial Y}}_{\equiv C_Y &gt; 0} \cdot \operatorname{dY} + \underbrace{\frac{\partial C(\cdot)}{\partial r}}_{\equiv C_r &lt; 0} \cdot \operatorname{dr} + \underbrace{\frac{\partial I(\cdot)}{\partial r}}_{\equiv I_r &lt; 0} \cdot \operatorname{dr} \therefore \operatorname{dY} - C_Y \cdot \operatorname{dY} = (C_r + I_r) \cdot \operatorname{dr} \Leftrightarrow \boxed{\boxed{\frac{\operatorname{dr}}{\operatorname{dY}}\bigg\vert_{IS'} = \frac{1-C_Y}{(C_r + I_r)} &lt; 0}}
$$
Since $|C_r+I_r| &gt; |I_r|$, the IS curve is now flatter and thus the income responds more to changes in the interest rate than before.