If $y=\frac{\sin (x+9)}{\cos x}$, then $\frac{d y}{d x}$ at $x=0$ is equal to

If $f(x)=1+x+\frac{x^2}{2}+\ldots+\frac{x^{100}}{100}$, then $f^{\prime}(1)$ is equal to

If $f(x)=\frac{x^n-a^n}{x-a}$ for some constant $a$, then $f^{\prime}(a)$ is equal to

If $f(x)=x^{100}+x^{99}+\ldots+x+1$, then $f^{\prime}(1)$ is equal to

If $f(x)=1-x+x^2-x^3+\ldots-x^{99}+x^{100}$, then $f^{\prime}(1)$ is equal to