IB Mathematics AA HL Flashcards -Derivatives of secx , cscx , cotx

In topic 5.6, we saw given derivatives for \( a^x \), \( \sin x \), \( \cos x \), \( e^x \) & \( \ln x \). Now we will simply find out derivatives for nine further functions and, therefore, nine new integrals as well.

RULES
\[
f(x) = \tan x \rightarrow f'(x) = \sec^2 x \quad \text{(ALL IN F.B.)}
\]
\[
f(x) = \sec x \rightarrow f'(x) = \sec x \tan x \quad \text{(to get this)}
\]
\[
f(x) = \cot x \rightarrow f'(x) = -\csc^2 x \quad \text{(to get this)}
\]
\[
f(x) = \log_a x \rightarrow f'(x) = \frac{1}{x \ln a}
\]
\[
f(x) = \sin x \rightarrow f'(x) = \cos x
\]
\[
f(x) = \cos x \rightarrow f'(x) = -\sin x
\]
\[
f(x) = \tan^{-1} x \rightarrow f'(x) = \frac{1}{1 + x^2}
\]

[q]

LINEAR \((ax+b)\) COMPOSITE INTEGRALS
In 5.10, we saw this extra aspect to basic integrals. The theory, briefly, is that when you differentiate \( f(ax+b) \), you get \( f'(ax+b) \cdot a \). So when you integrate a function with \( ax+b \) inside, you must divide by \( a \). This can be explained much better through the technique of integration by substitution:
\[
\text{If } f(f(x) = F(x) + c, \text{ then } \int f(ax+b) \, dx = \frac{1}{a}F(ax+b) + c
\]
(NOT IN F.B.)

[a]

INTEGRATING WITH PARTIAL FRACTIONS
You might see an integral in the form:
\[
\int \frac{1}{ax^2 + bx + c} \, dx
\]
You might be able to rewrite it as:
\[
\int \frac{1}{(ax+b)} \, dx, \text{ then use the arctan} \left( \frac{1}{\text{(tan)}} \text{ rule above.}
\]
This may not be possible, so another technique is to split it up into partial fractions, then integrate. See (???) for more details.

[x] Exit text