Differential-Calculus-Solver

Targeting the Derivative Calculus Domain

Derivative Rules

Basic Rules:

1. Constant Rule: $\frac{d}{dx}(c) = 0$
2. Power Rule: $\frac{d}{dx}(x^n) = nx^{n-1}$
3. Sum Rule: $\frac{d}{dx}(f(x) + g(x)) = \frac{d}{dx}(f(x)) + \frac{d}{dx}(g(x))$.
4. Difference Rule: $\frac{d}{dx}(f(x) - g(x)) = \frac{d}{dx}(f(x)) - \frac{d}{dx}(g(x))$.
5. Product Rule: $\frac{d}{dx}(f(x) \cdot g(x)) = f'(x) \cdot g(x) + f(x) \cdot g'(x)$.
6. Quotient Rule: $\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{(g(x))^2}$.

Exponential and Logarithmic Functions:

7. Exponential Function Euler Constant: $\frac{d}{dx}(e^x) = e^x$.
8. Natural Logarithm: $\frac{d}{dx}(\ln(x)) = \frac{1}{x}$.

Trigonometric Functions:

9. Sine Function: $\frac{d}{dx}(\sin(x)) = \cos(x)$.
10. Cosine Function: $\frac{d}{dx}(\cos(x)) = -\sin(x)$.
11. Tangent Function: $\frac{d}{dx}(\tan(x)) = \sec^2(x)$.
12. Secant Function: $\frac{d}{dx}(\sec(x)) = \sec^2(x)\tan(x)$.
13. Cosecant Function: $\frac{d}{dx}(\csc(x)) = -\csc(x)\cot(x)$.
14. Cotangent Function: $\frac{d}{dx}(\cot(x)) = -\csc^2(x)$.

Inverse Trigonometric Functions:

15. Arcsine Function: $\frac{d}{dx}(\arcsin(x)) = \frac{1}{\sqrt{1-x^2}}$.
16. Arccosine Function: $\frac{d}{dx}(\arccos(x)) = -\frac{1}{\sqrt{1-x^2}}$.
17. Arctangent Function: $\frac{d}{dx}(\arctan(x)) = \frac{1}{1+x^2}$.

Inverse Trigonometric Functions:

Chain Rule:

18. Chain Rule: If $y = f(g(x))$, then $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$, let $u = g(x)$.

Grammar

# <expression> := group
#              | <unary>
#              | <binary>
#              | <terminal>

# <group>     := LPAREN <expression> RPAREN
#
# <unary>     := MINUS <expression>

# <binary>    := <expression> PLUS   <expression>
#              | <expression> MINUS  <expression>
#              | <expression> TIMES  <expression>
#              | <expression> DIVIDE <expression>
#              | <expression> POWER  <expression>
#
# <terminal>  := <variable>
#              | <number>
#              | <function>
#              | <constant>
#
# <variable>  := [a-zA-Z]+
# <number>    := d+
# <constant>  := sin | cos | tan | ...
# <function>  := <constant> group

Reference

https://www.eeweb.com/tools/calculus-derivatives-and-limits-reference-sheet/