It's quite common to bump into people who think it is, but Kannada is not a derivative, a simplification, a corruption, or in short, a daughter of Sanskrit. Based on etymological and grammatical considerations, linguists place Kannada and Sanskrit in two separate language families, viz., Dravidian and Indo-Aryan. If this is the case, why are people misinformed? What prompts even educated Kannadigas to wrongly claim that Sanskrit is the mother of Kannada?
There are four main reasons.
1. Idea of Sanskrit as the mega-matriarch
There is this idea that Sanskrit is the mother of all the languages of the world. Kannada's is a special case of this. The problem with this idea is that it has no scientific backing. Linguists have shown that it is impossible to derive (using laws primarily of sound change) every language in the world from Sanskrit. That is, it is impossible to propose simple transformation rules (such as old Kannada's p changing to h in modern Kannada) to show that every language is derived from Sanskrit. Nor is it possible in the case of Kannada. Every now and then appears a novice who gets excited about one or two words in Kannada, known not to be of Sanskrit origin, "appearing similar" to words in Sanskrit. He or she then makes the claim that it proves the genetic relationship between the two languages. But for such claims to hold any water, he or she has to show that it is a general rule - and that's impossible.
2. Confusing writing with language
Many people cannot differentiate writing from language. Driven by this misconception, they look at Kannada writing, see that there are a lot of Sanskrit or Sanskrit-based words, and conclude that the Kannada language itself must be a derivative of Sanskrit. They talk in fancy percentages: "50% of Kannada is Sanskrit", "60% of Kannada is Sanskrit", etc. Apart from the lack of statistical backing for the actual numbers, such claims are wrong for the simple reason that writing and speech are two different things. Especially in the case of Kannada, writing, an elite preoccupation, has deviated significantly from speech due to the very fact of over-usage of Sanskrit words. The percentage claims on the language as a whole would be valid if what applies to writing applied to speech too, but this is not the case. Kannadigas make fun of those who speak Kannada like it's written: that's what immigrants do after picking up cheap books claiming to teach spoken Kannada. Much of written Kannada is unintelligible to common Kannadiga, not because he or she is incapable of grasping the content, but because of excessive use of Sanskrit words. There is an ongoing Kannada language movement which aims to fix these problems and bring written Kannada closer to the spoken language (including coining words) so that the benefits of writing and the ability to contribute to it are available to one and all. This is a requirement in today's age of compulsory primary education, a concept new to every Indian language, not just Kannada.
3. Grammatical mistakes
There is the question of grammar, which is closely related to the above point. Kannada's grammatical tradition, right from Kavirajamarga (850 CE) up until a decade or so ago, has essentially followed Sanskrit's, basically because of the huge influence of Sanskrit on the initial grammarians and the fact that Kannada literature was also quite heavily Sanskritised in its earlier stages. Thus, if the Sanskrit grammarians wrote of seven vibhaktis, Kannada grammarians followed suit even though only three could be properly called so in Kannada, and even though, unlike in Sanskrit, no vibhakti pratyaya in Kannada denotes gender and number over and above the noun-verb relationship. If the Sanskrit grammarians talked of karakas, the Kannada grammarians followed suit even though the very concept was unnecessary - unlike in Sanskrit, the mapping between vibhakti and meaning is one-to-one in Kannada. If the Sanskrit grammarians described samasas based on whether the second, first, both, or neither of the two participating words are central to the new word, their Kannada counterparts copied them even though every Kannada samasa has the second word as the central one. And then, if the Sanskrit grammarians described Sanskrit sandhi rules, Kannada grammarians applied all of them to Kannada grammar although it isn't necessary at all. One could go on and on about this, but the point is - if one picks up any popular Kannada grammar book, one gets the idea that Kannada's grammar is derived from Sanskrit's. But really, this is only a case of bad grammar writing. Put differently, the unwritten Kannada grammar on people's tongues is very unlike Sanskrit's, but the existing written grammars of Kannada tell a different - and wrong - story. Thankfully, this is being rectified as we speak.
4. The question of content
There was very little original writing in Kannada until very recently. It would not be an exaggeration to say that a large part of the Kannada corpus is essentially the result of a vernacularisation of Sanskrit literature. This has led to the idea that Sanskrit is the mother of Kannada, although nobody adds the qualifier 'when it comes to literature'. But the fact is, although language acts as a carrier for content, the two cannot be equated. Thus, when we translate English content into Kannada, which we're doing quite a bit, English doesn't become the latest mother of Kannada. Also, original writing in Kannada is now very much on the rise, and this originality has named people for mothers and fathers. There is still a lot of the aforementioned vernacularisation process left to be completed, but there's little focus on it due to noise from people lacking understanding of what Sanskrit really has to offer. They forget content and think learning Sanskrit is like learning every Indian language including Kannada. But that is just a bad joke.