The difference between $\int$ and $d()$?

Question

I got these wrong on a test, where I was given what is inside $d()$:

• $d(\sin x) = -\cos x$
  
• $d(\cos x) = \sin x$
  
• $d(\tan x) = \ln(\sec x)$
  
• $d\left(\frac{1}{4}x^4\right) = \frac{1}{20}x^5$
  
  

I got them all wrong; I was under the assumption that this:

• $d(\sin x) = -\cos x$
  Meant the same as this:
  
  
• $\int(-\cos x)\ dx = \sin x + c$
  
  Was I right, and just lost points for not including the "$dx$" after "$-\cos x$", or do these two symbols mean something different?

Answer 1

Issues of forgetting $\mathrm{d}x$ aside, you seem to have mixed up what $\int$ and $\mathrm{d}$ refer to. You appear to have integrated to compute $\mathrm{d}()$, and differentiated to compute $\int()$. But the meaning of the symbols is the other way around; $\mathrm{d}()$ is differentiation, and $\int()$ is integration.

e.g. it's $\mathrm{d}(x^7) = 7x^6 \mathrm{d}x$ and $\int x^7 \mathrm{d}x = \frac{1}{8} x^8 + C$, not the other way around.

Answer 2

Actually the derivative of $\sin x$ is $\cos x$, and the derivative of $\cos x$ is $-\sin x$. It does not seem like the $dx$ is the most important mistake. I have a strong feeling you have swapped the operations of integration and derivation.