diff --git a/neural-networks-case-study.md b/neural-networks-case-study.md
index 19be2560..2bf64a7c 100644
--- a/neural-networks-case-study.md
+++ b/neural-networks-case-study.md
@@ -90,7 +90,7 @@ $$
 L_i = -\log\left(\frac{e^{f_{y_i}}}{ \sum_j e^{f_j} }\right)
 $$
 
-We can see that the Softmax classifier interprets every element of \\(f\\) as holding the (unnormalized) log probabilities of the three classes. We exponentiate these to get (unnormalized) probabilities, and then normalize them to get probabilites. Therefore, the expression inside the log is the normalized probability of the correct class. Note how this expression works: this quantity is always between 0 and 1. When the probability of the correct class is very small (near 0), the loss will go towards (postiive) infinity. Conversely, when the correct class probability goes towards 1, the loss will go towards zero because \\(log(1) = 0\\). Hence, the expression for \\(L_i\\) is low when the correct class probability is high, and it's very high when it is low. 
+We can see that the Softmax classifier interprets every element of \\(f\\) as holding the (unnormalized) log probabilities of the three classes. We exponentiate these to get (unnormalized) probabilities, and then normalize them to get probabilites. Therefore, the expression inside the log is the normalized probability of the correct class. Note how this expression works: this quantity is always between 0 and 1. When the probability of the correct class is very small (near 0), the loss will go towards (positive) infinity. Conversely, when the correct class probability goes towards 1, the loss will go towards zero because \\(log(1) = 0\\). Hence, the expression for \\(L_i\\) is low when the correct class probability is high, and it's very high when it is low. 
 
 Recall also that the full Softmax classifier loss is then defined as the average cross-entropy loss over the training examples and the regularization:
 
diff --git a/optimization-2.md b/optimization-2.md
index 0b213366..d7972301 100644
--- a/optimization-2.md
+++ b/optimization-2.md
@@ -43,7 +43,7 @@ $$
 \frac{df(x)}{dx} = \lim_{h\ \to 0} \frac{f(x + h) - f(x)}{h}
 $$
 
-A technical note is that the division sign on the left-hand sign is, unlike the division sign on the right-hand sign, not a division. Instead, this notation indicates that the operator \\(  \frac{d}{dx} \\) is being applied to the function \\(f\\), and returns a different function (the derivative). A nice way to think about the expression above is that when \\(h\\) is very small, then the function is well-approximated by a straight line, and the derivative is its slope. In other words, the derivative on each variable tells you the sensitivity of the whole expression on its value. For example, if \\(x = 4, y = -3\\) then \\(f(x,y) = -12\\) and the derivative on \\(x\\) \\(\frac{\partial f}{\partial x} = -3\\). This tells us that if we were to increase the value of this variable by a tiny amount, the effect on the whole expression would be to decrease it (due to the negative sign), and by three times that amount. This can be seen by rearranging the above equation ( \\( f(x + h) = f(x) + h \frac{df(x)}{dx} \\) ). Analogously, since \\(\frac{\partial f}{\partial y} = 4\\), we expect that increasing the value of \\(y\\) by some very small amount \\(h\\) would also increase the output of the function (due to the positive sign), and by \\(4h\\).
+A technical note is that the division sign on the left-hand side is, unlike the division sign on the right-hand side, not a division. Instead, this notation indicates that the operator \\(  \frac{d}{dx} \\) is being applied to the function \\(f\\), and returns a different function (the derivative). A nice way to think about the expression above is that when \\(h\\) is very small, then the function is well-approximated by a straight line, and the derivative is its slope. In other words, the derivative on each variable tells you the sensitivity of the whole expression on its value. For example, if \\(x = 4, y = -3\\) then \\(f(x,y) = -12\\) and the derivative on \\(x\\) \\(\frac{\partial f}{\partial x} = -3\\). This tells us that if we were to increase the value of this variable by a tiny amount, the effect on the whole expression would be to decrease it (due to the negative sign), and by three times that amount. This can be seen by rearranging the above equation ( \\( f(x + h) = f(x) + h \frac{df(x)}{dx} \\) ). Analogously, since \\(\frac{\partial f}{\partial y} = 4\\), we expect that increasing the value of \\(y\\) by some very small amount \\(h\\) would also increase the output of the function (due to the positive sign), and by \\(4h\\).
 
 > The derivative on each variable tells you the sensitivity of the whole expression on its value.