Q: Find the derivative of $f(x) = \sin x$ with respect to $x$ using first principles.

By definition, derivative is $\lim{h \to 0} \frac{\sin(x+h) - \sin x}{h}$. Using $\sin A - \sin B = 2 \cos\frac{A+B}{2}\sin\frac{A-B}{2}$, the limit simplifies to $\cos x$.

Question 1

Find the derivative of $f(x) = \sin x$ with respect to $x$ using first principles.

Accepted Answer

By definition, derivative is $\lim{h 	o 0} \frac{\sin(x+h) - \sin x}{h}$. Using $\sin A - \sin B = 2 \cos\frac{A+B}{2}\sin\frac{A-B}{2}$, the limit simplifies to $\cos x$.

Question 2

Find the derivative of $f(x) = \cos x$.

Accepted Answer

Using the first principle or quotient rule on $\cos x$, the derivative of $\cos x$ with respect to $x$ is $-\sin x$.

Question 3

Find the derivative of $f(x) = 3x^2 + 5x - 7$.

Accepted Answer

Using the power rule and sum rule, $\frac{d}{dx}(3x^2) + \frac{d}{dx}(5x) - \frac{d}{dx}(7) = 6x + 5 - 0 = 6x + 5$.

Question 4

Find the derivative of $f(x) = \log x$ with respect to $x$.

Accepted Answer

Using first principles for natural logarithmic function, the derivative of $\loge x$ with respect to $x$ is $\frac{1}{x}$.

Question 5

What is the derivative of $\frac{u}{v}$ with respect to $x$ (Quotient Rule)?

Accepted Answer

According to the quotient rule of differentiation, $\frac{d}{dx}\left(\frac{u}{v}ight) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}$, provided $v 
eq 0$.

Limits and Derivatives