- Warm-up-exercises
Determine the derivative of the functions
-
for all
.
Determine the derivative of the function
-
for all
.
Determine the derivative of the function
-
for all
.
Determine directly (without the use of derivation rules) the derivative of the function
-
at any point
.
Prove that the real absolute value
-
is not differentiable at the point zero.
Determine the derivative of the function
-
Prove that the derivative of a rational function is also a rational function.
Consider
and
.
Determine the derivative of the composite function
directly and by the chain rule.
Prove that a polynomial
has degree
(or it is
), if and only if the
-th derivative of
is the zero poynomial.
Let
-
be two differentiable functions and consider
-
![{\displaystyle {}h(x)=(g(f(x)))^{2}f(g(x))\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a7d2e942dbd9b845345106811d0526f0a1ff580)
a) Determine the derivative
from the derivatives of
and
.
b) Let now
-
Compute
in two ways, one directly from
and the other by the formula of part
.
Let
be a field and let
be a
-vector space. Prove that given two vectors
there exists exactly one affine-linear map
-
sucht that
and
.
Determine the affine-linear map
-
such that
and
.
- Hand-in-exercises
Determine the derivative of the function
-
where
is the set where the denominator does not vanish.
Determine the tangents to the graph of the function
,
which are parallel to
.
Let
-
![{\displaystyle {}f(x)={\frac {x^{2}+5x-2}{x+1}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c174a97aefab57c99b16e3e32a85b0ad505b16e)
and
-
![{\displaystyle {}g(y)={\frac {y-2}{y^{2}+3}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d78990efe61837d9bcec851cda0d2925d73d6c26)
Determine the derivative of the composite
-
![{\displaystyle {}h(x)=g(f(x))\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d31ce8b35f5a3094e618677c22eb8b33ca136c0e)
directly and by the chain rule.
Determine the affine-linear map
-
whose graph passes through the two points
and
.
Let
be a subset and let
-
be differentiable functions. Prove the formula
-
![{\displaystyle {}{\left(f_{1}\cdots f_{n}\right)}'=\sum _{i=1}^{n}f_{1}\cdots f_{i-1}f_{i}'f_{i+1}\cdots f_{n}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9deb41414133a030a79c570dda9f92aa3c6cfcd5)