Calculus is the study of  derivatives. These derivatives are brought about by bringing the method of first principles.The method of first principle gives the results of all the derivatives.We shall first introduce some of the basic formulas of the derivatives that will be help full in solving the problems of the derivatives using sum,product and quotient rule which are the three fundamental rules for solving calculus problems.

Free Calculus Problem Solver Step by Step

Results of derivatives of algebraic ,trigonometric,exponential,and logarithmic functions.
1.$\frac{\mathrm{d} (sinx)}{\mathrm{d} x}$=cosx
2.$\frac{\mathrm{d} (cosx)}{\mathrm{d} x}$= -sinx
3.$\frac{\mathrm{d} (tanx)}{\mathrm{d} x}$=sec2x
4.$\frac{\mathrm{d} (secx)}{\mathrm{d} x}$=secxtanx
5.$\frac{\mathrm{d} (cosecx)}{\mathrm{d} x}$=-cosecxcotx
6.$\frac{\mathrm{d} (cotx)}{\mathrm{d} x}$=-cosec2x
7.$\frac{\mathrm{d} (e^{ax})}{\mathrm{d} x}$ = aeax
8.$\frac{\mathrm{d} (logx)}{\mathrm{d} x}$=1/x
we shall use the above results in solving the problems based on calculus step by step method.
Example 1
Find the derivative of x2tanx
Here we shall take u=x2 and v=tanx.
Formula
$u\frac{\mathrm{d}v}{\mathrm{d} x}+v\frac{\mathrm{d} u}{\mathrm{d} x}$
Solution
Step 1 $x\frac{\mathrm{d}tanx}{\mathrm{d} x}+tanx\frac{\mathrm{d} x}{\mathrm{d} x}$
Step 2 The deriavtive of tanx is sec2x and the derivative of x is 1.
Step 3 x(sec2x)+tanx(1)
Answer  x(sec2x)+tanx.
Example 2
Find the derivative of sinx using the steps
Solution 
step 1 : y= sinx
step 2 : $ \frac{\mathrm{dy} }{\mathrm{d} x}$=$ lim \Delta x\rightarrow 0\frac{f(x+\Delta x)-f(x)}{\Delta x}$
step 3 : $ \frac{\mathrm{dy} }{\mathrm{d} x}$= $lim \Delta x\rightarrow 0\frac{sin(x+\Delta x)-sin(x)}{\Delta x}$
step 4: $ lim \Delta x\rightarrow 0\frac{2cos\frac{2x+\Delta x}{2}sin(\frac{\Delta x}{2})}{\Delta x}$
step 5:  $ \lim_{\Delta x\to 0} \cos\left(\frac{2x+\Delta x}{2} \right )* \frac{\lim_{\Delta x\to 0}\sin\left(\frac{\Delta x}{2} \right )}{\Delta{x}\2}$
step 6:  $cos(\frac{2x}{2})* 1 $
step 7: $ \frac{\mathrm{dy} }{\mathrm{d} x}$ = cosx
Answer : cosx
Example 3
Find the derivative of  y=cosx using steps.
Solution
step 1 : y= cosx
step 2 : $ \frac{\mathrm{dy} }{\mathrm{d} x}$=$ lim \Delta x\rightarrow 0\frac{f(x+\Delta x)-f(x)}{\Delta x}$
step 3 : $ \frac{\mathrm{dy} }{\mathrm{d} x}$= $lim \Delta x\rightarrow 0-\frac{cos(x+\Delta x)-cos(x)}{\Delta x}$
step 4: $ lim \Delta x\rightarrow 0\frac{2sin\frac{2x+\Delta x}{2}sin(\frac{\Delta x}{2})}{\Delta x}$
step 5:  $ \lim_{\Delta x\to 0} \cos\left(\frac{2x+\Delta x}{2} \right )* \frac{\lim_{\Delta x\to 0}\sin\left(\frac{\Delta x}{2} \right )}{\Delta{x}\2}$
step 6:  $cos(\frac{2x}{2})* 1 $
step 7: $ \frac{\mathrm{dy} }{\mathrm{d} x}$ = -sinx
Answer : -sinx
Example 4
Find the derivative of y= tan2x
Solution
Here we shall consider u=tanx
Step1 y=(u)
step2 This falls under chain rule
step3 we here adapt a formulae for chain rule.
step4 $\frac{\mathrm{d}y }{\mathrm{d} x}=\frac{\mathrm{d} u2}{\mathrm{d} u}*\frac{\mathrm{d} tanx}{\mathrm{d} x}$
step5 2u*sec2x
step6 2tanxsec2x
Answer: 2 tanx sec2x.