Differentiate. y=cotx to the power sinx + tan x to the power cos x
Answers
Answer:
d(sinx)
=cosx
\displaystyle\frac{{{d}{\left( \cos{{x}}\right)}}}{{\left.{d}{x}\right.}}=- \sin{{x}}
dx
d(cosx)
=−sinx
\displaystyle\frac{{{d}{\left( \tan{{x}}\right)}}}{{{\left.{d}{x}\right.}}}={{\sec}^{2}{x}}
dx
d(tanx)
=sec
2
x
Explore animations of these functions with their derivatives here:
Differentiation Interactive Applet - trigonometric functions.
In words, we would say:
The derivative of sin x is cos x,
The derivative of cos x is −sin x (note the negative sign!) and
The derivative of tan x is sec2x.
Now, if u = f(x) is a function of x, then by using the chain rule, we have:
\displaystyle\frac{{{d}{\left( \sin{{u}}\right)}}}{{{\left.{d}{x}\right.}}}= \cos{{u}}\frac{{{d}{u}}}{{{\left.{d}{x}\right.}}}
dx
d(sinu)
=cosu
dx
du
\displaystyle\frac{{{d}{\left( \cos{{u}}\right)}}}{{\left.{d}{x}\right.}}=- \sin{{u}}\frac{{{d}{u}}}{{{\left.{d}{x}\right.}}}
dx
d(cosu)
=−sinu
dx
du
\displaystyle\frac{{{d}{\left( \tan{{u}}\right)}}}{{{\left.{d}{x}\right.}}}={{\sec}^{2}{u}}\frac{{{d}{u}}}{{{\left.{d}{x}\right.}}}
dx
d(tanu)
=sec
2
u
dx
du
Example 1
Differentiate \displaystyle{y}= \sin{{\left({x}^{2}+{3}\right)}}y=sin(x
2
+3).
Answer
First, let: \displaystyle{u}={x}^{2}+{3}u=x
2
+3 and so \displaystyle{y}= \sin{{u}}y=sinu.
We have:
\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}=\frac{{{\left.{d}{y}\right.}}}{{{d}{u}}}\frac{{{d}{u}}}{{{\left.{d}{x}\right.}}}
dx
dy
=
du
dy
dx
du
\displaystyle= \cos{{u}}\frac{{{d}{u}}}{{{\left.{d}{x}\right.}}}=cosu
dx
du
\displaystyle= \cos{{\left({x}^{2}+{3}\right)}}\frac{{{d}{\left({x}^{2}+{3}\right)}}}{{{\left.{d}{x}\right.}}}=cos(x
2
+3)
dx
d(x
2
+3)
\displaystyle={2}{x}\ \cos{{\left({x}^{2}+{3}\right)}}=2x cos(x
2
+3)IMPORTANT:
cos x2 + 3
does not equal
cos(x2 + 3).
The brackets make a big difference. Many students have trouble with this.
Here are the graphs of y = cos x2 + 3 (in green) and y = cos(x2 + 3) (shown in blue).
The first one, y = cos x2 + 3, or y = (cos x2) + 3, means take the curve y = cos x2 and move it up by \displaystyle{3}3 units.
Graph y = cos(x^2+3)
The second one, y = cos(x2 + 3), means find the value (x2 + 3) first, then find the cosine of the result.
They are quite different!
Graph y = cos(x^2+3)