To get the best result from differential calculus playground, use the following guides to correctly form your calculus  expression

The differential coefficient of a sum or difference below

3x^2 + 2x - 7
5x^4 - 4x^3 + 2x^2
8x^3 - 6x^2 + 4x - 1
x^4 + 3x^3 - 2x^2 + 5x
4x^5 - 2x^4 + 6x^3 - 3x^2 + x

can be written as

3x^2 + 2x - 7.
5x^4 - 4x^3 + 2x^2.
8x^3 - 6x^2 + 4x - 1.
 x^4 + 3x^3 - 2x^2 + 5x.
4x^5 - 2x^4 + 6x^3 - 3x^2 + x.

If you need to perform an operation other than addition, such as 3x^2 + 2x - 7, and would like to experiment with the order of operations involving exponents, click on button 2, and continue from there.

The differential of exponential functions

1. Find the derivative of the function f(x) = e^{3x}.
2. Calculate the derivative of g(x) = 2e^{-4x}.
3. Determine the derivative of h(x) = 5e^{2x} + 3e^{-x}.
4. Find the derivative of y = ae^{bx}, where a and b are constants.
5. Calculate the derivative of f(t) = e^{kt} - 2e^{-2t}, where k is a constant.
6. Determine the derivative of g(x) = e^{2x} \cdot e^{-3x}.
7. Find the derivative of h(t) = 4e^{3t} - 2e^{5t}.
8. Calculate the derivative of y = e^{2x} \cdot \sin(x).
9. Determine the derivative of f(x) = 3e^x + 5e^{2x}.
10. Find the derivative of g(t) = e^{kt} \cdot \cos(t), where k is a constant.

can be written into the playground box as

1. e^(3x)
2. 2e^(-4x)
3. 5e^(2x) + 3e^(-x)
4. ae^(bx)
5. e^(kt) - 2e^(-2t)
6. e^(2x) cdot e^(-3x)
7. 4e^(3t) - 2e^(5t)
8. e^(2x) cdot sin(x)
9. 3e^x + 5e^(2x)
10. e^(kt) cdot cos(t)

The differentiation of logarithmic functions

1. Find the derivative of the function f(x) = \ln(x^2 + 1).
2. Calculate the derivative of g(x) = \ln(2x + 3).
3. Determine the derivative of h(x) = \ln(\sqrt{x} + 2).
4. What is the derivative of the function j(x) = \ln(e^{2x})?
5. Find the derivative of the function k(x) = \ln(\sin(x) + 1).
6. Calculate the derivative of m(x) = \ln(\cos(3x) + 2).
7. Determine the derivative of n(x) = \ln(5^x + 3).
8. What is the derivative of the function p(x) = \ln(\frac{1}{x} + e^x)?
9. Find the derivative of the function q(x) = \ln(\tan(4x) + 5).
10. Calculate the derivative of the function r(x) = \ln(x^3 + 4x^2 + 2x + 1).

can be written as

1. ln(x^2 + 1)
2. ln(2x + 3)
3. ln(sqrt(x) + 2)
4. ln(e^(2x))
5. ln(sin(x) + 1)
6. ln(cos(3x) + 2)
7. ln(5^x + 3)
8. ln(1/x + e^x)
9. ln(tan(4x) + 5)
10. ln(x^3 + 4x^2 + 2x + 1)

The differentiation of function of a function

1. Find the derivative of f(x) = \sin(2x) using the chain rule.
2. Calculate the derivative of g(t) = \ln(3t^2) with respect to t.
3. Determine the derivative of h(x) = \cos(4x^2) using the chain rule.
4. Find the derivative of y(t) = e^{3t^3} with respect to t.
5. Calculate the derivative of z(x) = \sqrt{5x + 1} using the chain rule.
6. Determine the derivative of v(t) = \tan(2t) with respect to t.
7. Find the derivative of u(x) = \frac{1}{2x} using the chain rule.
8. Calculate the derivative of p(t) = \sin(3t^2) with respect to t.
9. Determine the derivative of q(x) = e^{4x^3} using the chain rule.
10. Find the derivative of r(t) = \ln(2t + 3) with respect to t.

can be written as

1. sin(2x)
2. ln(3t^2)
3. cos(4x^2)
4. e^(3t^3)
5. sqrt(5x + 1)
6. tan(2t)
7. (1)/(2x)
8. sin(3t^2)
9. e^(4x^3)
10. ln(2t + 3)

The differentiation of a product

1. Differentiate f(x) = x^2 \cdot \sin(x) with respect to x.
2. Find the derivative of g(x) = e^{2x} \cdot \cos(x).
3. Calculate the derivative of h(x) = \sqrt{x} \cdot \ln(x).
4. Determine the differentiation of p(x) = x^3 \cdot \tan(x).
5. Find the derivative of q(x) = \dfrac{\cos(x)}{x^2}.
6. Differentiate r(x) = e^x \cdot \sec(x) with respect to x.
7. Calculate the derivative of s(x) = x \cdot \sin(2x).
8. Determine the differentiation of t(x) = \ln(x) \cdot \csc(x).
9. Find the derivative of u(x) = \dfrac{x^2}{\sin(x)}.
10. Differentiate v(x) = e^{3x} \cdot \cot(x) with respect to x.

written as

1. x^2 * sin(x)
2. e^(2x) * cos(x)
3. sqrt(x) * ln(x)
4. x^3 * tan(x)
5. cos(x) / x^2
6. e^x * sec(x)
7. x * sin(2x)
8. ln(x) * csc(x)
9. x^2 / sin(x)
10. e^(3x) * cot(x)

The differentiation of quotient

1. What is the derivative of the function \frac{2x^3 + 4}{x^2 - 3x + 2}
2. Compute the derivative of \frac{5x^2 + 3x - 7}{2x^3 - 5x + 1}.
3. Find the derivative of \frac{e^{2x} + x^2}{3x - 5}.
4. Determine the derivative of \frac{\sin^2(x) - \cos(x)}{x^2 + 1}.
5. Calculate the derivative of \frac{3x^4 + 2x^3 - x}{4x^2 - 1}.
6. What is the derivative of \frac{\ln(x) + e^x}{x^2 - 1}?
7. Compute the derivative of \frac{2x^2 - x + 1}{x^3 + x^2 - x + 1}.
8. Find the derivative of \frac{\sqrt{x} + x^3}{2x + \cos(x)}.
9. Determine the derivative of \frac{3x^3 - 4x}{\sqrt{x} + 1}.
10. Calculate the derivative of \frac{\tan(x) + \sin(x)}{x^2 - x + 1}.

can be written as

(2x^3 + 4)/(x^2 - 3x + 2)
(5x^2 + 3x - 7)/(2x^3 - 5x + 1)
(e^(2x) + x^2)/(3x - 5)
(sin^2(x) - cos(x))/(x^2 + 1)
(3x^4 + 2x^3 - x)/(4x^2 - 1)
(ln(x) + e^x)/(x^2 - 1)
(2x^2 - x + 1)/(x^3 + x^2 - x + 1)
(sqrt(x) + x^3)/(2x + cos(x))
(3x^3 - 4x)/(sqrt(x) + 1)
(tan(x) + sin(x))/(x^2 - x + 1)

About Differential Calculus Playground

The "Differential Calculus Playground" is a valuable educational tool that allows users to test their knowledge of differential calculus. This platform provides quick answers and visual graphs for the following topics: differentiation of common functions, differentiation of products and quotients, application of differentiation in various scenarios such as rates of change, turning points, maximum and minimum values, tangents, and normals. It also covers differentiation of parametric equations, implicit functions, logarithmic functions, hyperbolic functions, and inverse trigonometric functions. This tool is essential for students and enthusiasts looking to strengthen their understanding of calculus concepts.

Question 1:
Find the derivative of f(x) = 3x^2 - 2x + 5 from first principles.

Solution 1:
Given function: f(x) = 3x^2 - 2x + 5

To find the derivative from first principles, we use the definition of derivative:

f'(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h}

Substitute f(x) into the formula:

f'(x) = \lim_{{h \to 0}} \frac{3(x+h)^2 - 2(x+h) + 5 - (3x^2 - 2x + 5)}{h}

Expand and simplify the expression:

f'(x) = \lim_{{h \to 0}} \frac{3(x^2 + 2xh + h^2) - 2x - 2h + 5 - 3x^2 + 2x - 5}{h}

f'(x) = \lim_{{h \to 0}} \frac{3x^2 + 6xh + 3h^2 - 2x - 2h + 5 - 3x^2 + 2x - 5}{h}

f'(x) = \lim_{{h \to 0}} \frac{6xh + 3h^2 - 2h}{h}

f'(x) = \lim_{{h \to 0}} 6x + 3h - 2

f'(x) = 6x - 2

Therefore, the derivative of f(x) = 3x^2 - 2x + 5 is f'(x) = 6x - 2.

Question 2:
Find the derivative of g(x) = 4x^3 + 2x^2 - x from first principles.

Solution 2:
Given function: g(x) = 4x^3 + 2x^2 - x

To find the derivative from first principles, we use the definition of derivative:

g'(x) = \lim_{{h \to 0}} \frac{g(x+h) - g(x)}{h}

Substitute g(x) into the formula:

g'(x) = \lim_{{h \to 0}} \frac{4(x+h)^3 + 2(x+h)^2 - (x+h) - (4x^3 + 2x^2 - x)}{h}

Expand and simplify the expression:

g'(x) = \lim_{{h \to 0}} \frac{4(x^3 + 3x^2h + 3xh^2 + h^3) + 2(x^2 + 2xh + h^2) - x - h - 4x^3 - 2x^2 + x}{h}
g'(x) = \lim_{{h \to 0}} \frac{4x^3 + 12x^2h + 12xh^2 + 4h^3 + 2x^2 + 4xh + 2h^2 - x - h - 4x^3 - 2x^2 + x}{h}
g'(x) = \lim_{{h \to 0}} \frac{12x^2h + 12xh^2 + 4h^3 + 4xh + 2h^2 - h}{h}
g'(x) = \lim_{{h \to 0}} 12x^2 + 12xh + 4h^2 + 4x + 2h - 1
g'(x) = 12x^2 + 4x - 1

Therefore, the derivative of g(x) = 4x^3 + 2x^2 - x is

g'(x) = 12x^2 + 4x - 1

Question 3:
Find the derivative of h(x) = sin(x) from first principles.

Solution 3:
Given function: h(x) = \sin(x)

To find the derivative from first principles, we use the definition of derivative:

h'(x) = \lim_{{h \to 0}} \frac{h(x+h) - h(x)}{h}

Substitute h(x) into the formula:

h'(x) = \lim_{{h \to 0}} \frac{\sin(x+h) - \sin(x)}{h}

Apply the angle sum identity for sine function:

h'(x) = \lim_{{h \to 0}} \frac{\sin(x)\cos(h) + \cos(x)\sin(h) - \sin(x)}{h}

Simplify the expression:

h'(x) = \lim_{{h \to 0}} \frac{\sin(x)\cos(h) + \cos(x)\sin(h) - \sin(x)}{h}
h'(x) = \lim_{{h \to 0}} \frac{\sin(x)(\cos(h) - 1) + \cos(x)\sin(h)}{h}
h'(x) = \lim_{{h \to 0}} \frac{\sin(x)(\cos(h) - 1)}{h} + \lim_{{h \to 0}} \frac{\cos(x)\sin(h)}{h}
h'(x) = \sin(x)\lim_{{h \to 0}} \frac{\cos(h) - 1}{h} + \cos(x)\lim_{{h \to 0}} \frac{\sin(h)}{h}

Using the limit definition of derivative for sine function, we know:
\lim_{{h \to 0}} \frac{\sin(h)}{h} = 1

Therefore, the derivative of h(x) = sin(x) is h'(x) = cos(x).

Question 4:
Find the derivative of k(x) = \sqrt{x} from first principles.

Solution 4:
Given function: k(x) = \sqrt{x}

To find the derivative from first principles, we use the definition of derivative:
k'(x) = \lim_{{h \to 0}} \frac{k(x+h) - k(x)}{h}

Substitute k(x) into the formula:
k'(x) = \lim_{{h \to 0}} \frac{\sqrt{x+h} - \sqrt{x}}{h}

Rationalize the numerator:
k'(x) = \lim_{{h \to 0}} \frac{(\sqrt{x+h} - \sqrt{x})(\sqrt{x+h} + \sqrt{x})}{h(\sqrt{x+h} + \sqrt{x})}
k'(x) = \lim_{{h \to 0}} \frac{(x+h) - x}{h(\sqrt{x+h} + \sqrt{x})}
k'(x) = \lim_{{h \to 0}} \frac{h}{h(\sqrt{x+h} + \sqrt{x})}
k'(x) = \lim_{{h \to 0}} \frac{1}{\sqrt{x+h} + \sqrt{x}}
k'(x) = \frac{1}{2\sqrt{x}}

Therefore, the derivative of k(x) = √x is k'(x) = 1/(2√x).

Question 5:
Find the derivative of y(x) = e^xfrom first principles.

Solution 5:
Given function: y(x) = e^x

To find the derivative from first principles, we use the definition of derivative:
y'(x) = \lim_{{h \to 0}} \frac{y(x+h) - y(x)}{h}

Substitute y(x) into the formula:
y'(x) = \lim_{{h \to 0}} \frac{e^{x+h} - e^x}{h}

Apply the properties of exponential functions:
y'(x) = \lim_{{h \to 0}} \frac{e^x \cdot e^h - e^x}{h}
y'(x) = \lim_{{h \to 0}} \frac{e^x (e^h - 1)}{h}

Since \lim_{{h \to 0}} \frac{e^h - 1}{h} = 1,
y'(x) = e^x

Therefore, the derivative of y(x) = e^x is y'(x) = e^x.

Questions

1. \frac{d}{dx}(f(x)g(x))

2. \frac{d}{dx}(3x^2 \cdot \sin(x))

3. \frac{d}{dx}(e^x \cdot \cos(x))

4. \frac{d}{dx}(2x \cdot \ln(x))

5. \frac{d}{dx}(x^3 \cdot e^{2x})

6. \frac{d}{dx}(4x \cdot \tan(x))

7. \frac{d}{dx}(6x^2 \cdot \sqrt{x})

8. \frac{d}{dx}(\ln(x) \cdot e^x)

9. \frac{d}{dx}(2x^3 \cdot \sin(2x))

10. \frac{d}{dx}(x \cdot \cos(x))

11. \frac{d}{dx}(4x^2 \cdot e^x)

12. \frac{d}{dx}(e^{3x} \cdot \tan(x))

13. \frac{d}{dx}(5x \cdot \ln(2x))

14. \frac{d}{dx}(x^4 \cdot \cosh(x))

15. \frac{d}{dx}(e^{-x} \cdot \sin(x))

Answer:

1. g(x)+f(x)g'(x)

2. 6x\sin(x)+3x^2\cos(x)

3. e^x\cos(x)-e^x\sin(x)

4. 2\ln(x)+2

5. 3x^2e^{2x}+2x^3e^{2x}

6. 4\tan(x)+4x\sec^2(x)

7. 12x\sqrt{x}+3x^2\sqrt{x}

8. \frac{1}{x}e^x+\ln(x)e^x

9. 6x^2\sin(2x)+4x^3\cos(2x)

10. \cos(x)-x\sin(x)

11. 4x^2e^x+8xe^x

12. 3e^{3x}\tan(x)+e^{3x}\sec^2(x)

13. 5\ln(2x)+5 [\frac{1}{x}]

14. 4x^3\cosh(x)+x^4\sinh(x)

15. -e^{-x}\sin(x)-e^{-x}\cos(x)