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Wize University Calculus 1 Textbook > Derivatives

Differentiation Laws

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Differentiation Laws

Let ff and gg be two differentiable functions and kk be a constant. We have the following rules:

  1. Sum Rule: ddx(f(x)+g(x))=dfdx+dgdx\dfrac{d}{dx}\Big(f(x)+g(x)\Big)=\dfrac{df}{dx}+\dfrac{dg}{dx}
  2. Difference Rule: ddx(f(x)−g(x))=dfdx−dgdx\dfrac{d}{dx}\Big(f(x)-g(x)\Big)=\dfrac{df}{dx}-\dfrac{dg}{dx}
  3. Constant Multiple Rule: ddx(kf(x))=kdfdx\dfrac{d}{dx}\Big(kf(x)\Big)=k\dfrac{df}{dx}
Extra Practice
Derivative of Polynomials: Product Rule
Find the derivative of f(x)=(x3/2+2x)(x7/4+x)f(x)=\left(x^{3/2}+\frac{2}{x}\right)\left(x^{7/4}+x\right)
Differentiation laws
What is the derivative of the function f(x)=x3+1x2f(x)=\frac{x^3+1}{x^2}?
Differentiation: Piecewise Differentiable Function
Find AA and BB for which f(x)f\left(x\right) is differentiable everywhere. Answers are in the form (A,B)\left(A,B\right)
f(x)={xex2+1, if x≥1Ax+B, if x<1f(x)=\begin{cases} xe^{x^2+1}, \text{ if } x\geq 1 \\ Ax+B, \text{ if } x < 1 \end{cases}
Differentiation: Piecewise Differentiable Function
Find AA and BB for which f(x)f\left(x\right) is differentiable everywhere.
f(x)={xex2+1, if x≥1Ax+B, if x<1f(x)=\begin{cases} xe^{x^2+1}, \text{ if } x\geq 1 \\ Ax+B, \text{ if } x < 1 \end{cases}
Derivative of Polynomials: Product Rule
Find the derivative of f(x)=(x3/2+2x)(x7/4+x)f(x)=\left(x^{3/2}+\frac{2}{x}\right)\left(x^{7/4}+x\right)
Differentiation: Piecewise Differentiable Function
Find AA and BB for which f(x)f\left(x\right) is differentiable everywhere.
f(x)={xex2+1, if x≥1Ax+B, if x<1f(x)=\begin{cases} xe^{x^2+1}, \text{ if } x\geq 1 \\ Ax+B, \text{ if } x < 1 \end{cases}
Differentiation: Piecewise Differentiable Function
Find AA and BB for which f(x)f\left(x\right) is differentiable everywhere.
f(x)={xex2+1, if x≥1Ax+B, if x<1f(x)=\begin{cases} xe^{x^2+1}, \text{ if } x\geq 1 \\ Ax+B, \text{ if } x < 1 \end{cases}
Differentiation: Piecewise Differentiable Function
Find AA and BB for which f(x)f\left(x\right) is differentiable everywhere.
f(x)={xex2+1, if x≥1Ax+B, if x<1f(x)=\begin{cases} xe^{x^2+1}, \text{ if } x\geq 1 \\ Ax+B, \text{ if } x < 1 \end{cases}
Derivatives: Logarithmic Functions
Compute the derivative of f(x)=xx+1f(x) = x^{x + 1}. Remember that log⁡x=log⁡ex=ln⁡x\log x = \log_e x = \ln x.
The chain rule
Find the derivative of h(x)=log⁡(cos⁡(x))h(x) = \log(\cos(x)). Remember that log⁡x=log⁡ex=ln⁡x\log x = \log_e x = \ln x.
The Quotient Rule
Find the derivative of g(x)=x2−52x+1g(x) = \frac{x^2 - 5}{2x + 1}
The product rule
Find the derivative of f(x)=x3exf(x) = x^3 e^x
Differentiation: Piecewise Differentiable Function
Find AA and BB for which f(x)f\left(x\right) is differentiable everywhere.
f(x)={xex2+1, if x≥1Ax+B, if x<1f(x)=\begin{cases} xe^{x^2+1}, \text{ if } x\geq 1 \\ Ax+B, \text{ if } x < 1 \end{cases}
Derivative of Polynomials: Product Rule
Find the derivative of f(x)=(x3/2+2x)(x7/4+x)f(x)=\left(x^{3/2}+\frac{2}{x}\right)\left(x^{7/4}+x\right)
Differentiation laws
What is the derivative of the function f(x)=x3+1x2f(x)=\frac{x^3+1}{x^2}?
Is f(x)f(x) differentiable at x=1x=1 ? If so, find f′(1)f'(1).
f(x)={x+1if x<112x2+32if x≥1 f(x) = \begin{cases} x+1 & \text{if } x < 1 \\ \frac{1}{2}x^2 + \frac{3}{2} & \text{if } x \geq 1 \end{cases}
Differential Laws: nth Derivatives
If y=(10e+1)10y=\left(10e+1\right)^{10}, find the 9th derivative of yy.
If f(x)f(x) is differentiable everywhere, find AA and BB.
f(x)={x2+1     if x≥0Ax+B       if  x<0f(x)=\left\{ \begin{array}{ll} \displaystyle x^2+1\quad\quad\,\,\,\,\,\text{if}\,x\geq 0 \\ Ax+B\quad\,\,\,\,\,\,\,\text{if}\,\, x<0\\ \end{array} \right.