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CalculusPop offers a wide range of professional solutions to differential calculus problems for free.
Example: \( 5x^4 + 3x^2 – 7x + 9 \)
Rule: Power Rule \( \frac{d}{dx} x^n = n x^{n-1} \)
The tool can differentiate terms with integer or fractional powers of \(x\).
Example: \( 7 \), \( -3 \)
Rule: Constant Rule \( \frac{d}{dx} C = 0 \)
The tool correctly identifies constants and returns 0 as their derivative.
Example: \( \frac{1}{x^2} \), \( \frac{5}{2x^3} \)
Rule: Quotient Rule \( \frac{d}{dx} \frac{u}{v} = \frac{u’v – uv’}{v^2} \)
It handles fractional terms like \( \frac{1}{x^n} \), rewriting them as \( x^{-n} \) or applying the quotient rule.
Example: \( (3x^2)(\sin(x)) \), \( (e^x)(x^3) \)
Rule: Product Rule \( \frac{d}{dx} (u \cdot v) = u’v + uv’ \)
The tool applies the product rule for terms that are products of two functions.
Example: \( \frac{x^3}{\sin(x)} \), \( \frac{e^x}{x} \)
Rule: Quotient Rule \( \frac{d}{dx} \frac{u}{v} = \frac{u’v – uv’}{v^2} \)
The tool detects division of functions and applies the quotient rule appropriately.
Example: \( \sqrt{x} \), \( x^{1/2} \), \( \frac{1}{\sqrt{x}} \)
Rule: Power Rule \( \frac{d}{dx} x^n = n x^{n-1} \)
The tool handles square roots by converting them to fractional powers \(x^{1/2}\) and differentiating using the power rule.
Example: \( e^x \), \( e^{2x} \)
Rule: Chain Rule \( \frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x) \)
The tool differentiates exponential functions like \(e^x\), \(e^{kx}\), and handles more complex exponential expressions.
Example: \( \ln(x) \), \( \ln(3x^2) \)
Rule: Chain Rule \( \frac{d}{dx} \ln(x) = \frac{1}{x} \)
The tool correctly differentiates natural logarithmic expressions.
Example: \( \sin(x) \), \( \cos(x) \), \( \tan(x) \), \( \csc(x) \), \( \sec(x) \), \( \cot(x) \)
Rule: Chain Rule (differentiation of composite functions)
Derivatives of common trigonometric functions:
The tool can differentiate trigonometric functions, including those nested inside other functions.
Example: \( \sin(3x^2) \), \( e^{x^2} \), \( \ln(\sqrt{x}) \)
Rule: Chain Rule \( \frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x) \)
The tool can handle functions inside other functions, applying the chain rule for nested functions.
Example: \( \sinh(x) \), \( \cosh(x) \)
Rule: Chain Rule
The tool differentiates hyperbolic functions:
\( \frac{d}{dx} \sinh(x) = \cosh(x) \)
\( \frac{d}{dx} \cosh(x) = \sinh(x) \)
Example: \( x^n \) where \(n\) is any real number, including negative or fractional powers.
Rule: Power Rule
The tool can handle higher-order powers and fractional powers, applying the power rule accordingly.
Example: \( \arcsin(x) \), \( \arccos(x) \), \( \arctan(x) \)
Rule: Chain Rule
The tool handles inverse trigonometric functions like:
\( \frac{d}{dx} \arcsin(x) = \frac{1}{\sqrt{1 – x^2}} \)
\( \frac{d}{dx} \arctan(x) = \frac{1}{1 + x^2} \)
The derivative of a constant is always zero.
For any power of \( x \), \( \frac{d}{dx}(x^n) = nx^{n-1} \).
The derivative of a sum or difference of functions is the sum or difference of their derivatives.
Example: \( \frac{d}{dx}(x^2 + 3x – 1) = 2x + 3 \).
For two functions \( u(x) \) and \( v(x) \), the derivative is given by:
\[
\frac{d}{dx}[u \cdot v] = u \cdot \frac{dv}{dx} + v \cdot \frac{du}{dx}
\]
Example: \( \frac{d}{dx}(x \cdot e^x) = x \cdot e^x + e^x \).
For a quotient of two functions \( u(x) \) and \( v(x) \), the derivative is:
\[
\frac{d}{dx}\left( \frac{u}{v} \right) = \frac{v \cdot \frac{du}{dx} – u \cdot \frac{dv}{dx}}{v^2}
\]
Example: \( \frac{d}{dx}\left( \frac{2x}{x+1} \right) = \frac{(x+1)(2) – 2x(1)}{(x+1)^2} \).
For a composite function \( f(g(x)) \), the chain rule states:
\[
\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)
\]
Example: \( \frac{d}{dx}(\sin(2x)) = \cos(2x) \cdot 2 \).
Sine: \( \frac{d}{dx}(\sin(x)) = \cos(x) \)
Cosine: \( \frac{d}{dx}(\cos(x)) = -\sin(x) \)
Tangent: \( \frac{d}{dx}(\tan(x)) = \sec^2(x) \)
For any exponential function \( e^x \), the derivative is:
\[
\frac{d}{dx}(e^x) = e^x
\]
Example: \( \frac{d}{dx}(e^{2x}) = 2e^{2x} \).
For the natural logarithm, the derivative is:
\[
\frac{d}{dx}(\log(x)) = \frac{1}{x}
\]
Example: \( \frac{d}{dx}(\log(x^2)) = \frac{2}{x} \).
The solver can also calculate higher-order derivatives (e.g., the second derivative, third derivative, etc.).
Example: The second derivative of \( x^3 \) is \( \frac{d^2}{dx^2}(x^3) = 6x \).
For functions involving square roots or other fractional exponents:
\[
\frac{d}{dx}(x^{1/2}) = \frac{1}{2}x^{-1/2}
\]
Example: \( \frac{d}{dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}} \).
Arcsin: \( \frac{d}{dx}(\sin^{-1}(x)) = \frac{1}{\sqrt{1 – x^2}} \)
Arccos: \( \frac{d}{dx}(\cos^{-1}(x)) = \frac{-1}{\sqrt{1 – x^2}} \)
Arctan: \( \frac{d}{dx}(\tan^{-1}(x)) = \frac{1}{1 + x^2} \)
Solve integral calculus using integration techniques, including algebraic and trigonometric substitution methods
CalculusPop offers customized and sophisticated solutions for differential and integral calculus problems, utilizing cutting-edge technology to handle even the most complex calculations in real-time. CalculusPop services are free of charge.
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